Okay, I did not find a mention of a last will of Richard Cameron expressing anything about a direct appointment of House Davion to the position of First Lord, should Cameron be killed.
However...on Page 58 of the PDF'd House Davion, there is the following:
"Davion tradition has it that prior to the execution of Richard Cameron, the First Lord named John Cameron as Regent for his two year old daughter Amanda. Certainly, House Davion would later use tradition as the key to its claim as rightful heir to the post of First Lord, while none of the other Houses could advance anything so significant on their own behalf."
As to the topic...
Hard to call overall. If you take out "the writer force" behind any topic and weigh completely by what seems reasonable, then these are changes/outlines as I would see them:
Hegemony Invasion:
First, Amaris is likely to retaliate if Davion overtly supports Kerensky. That much is a given as Seawolf has pointed out. Amaris is not hard to call in doing this as he is pretty much the archetypal despot.
Second, based on the above, Amaris killing off Davionist peoples/supporters. This might rally the Davion people more for the cause other than simple fiat by John Davion.
Result: Amaris is still doomed, it probably just happens faster.
Post-Amaris League Actions, Succession Wars, Etc.:
Were Davion to throw support behind Kerensky (wouldn't happen, but ok) I see the following:
The High Council would still probably break up, temporarily. One Successor State going to with the Regular Army means the balance tips toward order, not chaos in the short term. This is simple reasoning because the House Lords, every one of them, is playing the game to amass power - not simply go to war. Why?
Going to war and amassing power are two distinct issues and one does not equal the other. In many cases, going to war is the wrong way to amass power because it is always a more efficient thing to have power bestowed by a greatful populace than to take it by force. Force means anger and anger robs power unless fear is used to counter the anger, which rarely works for long anyway.
However, the House's belligerancy being what it was, the Succession War was inevitable because no one could agree on a compromise choice for First Lord. Thus, Kerensky left and the bottom fell out of any support for a Star League.
But, if Davion or another House were to declare for the Regular Army in some fashion as to make a Star League with whatever concessions were required things change. Of course, a House would never willingly commit political suicide, but for the sake of discussion, let's assume House Davion does.
When the other Great Houses see this, it will affect their political movments as much as a stone dropped in a pond would affect all its borders. So, assuming Davion declares for the League, the following conclusions are reached, at least in my view:
A. House Kurita: The easiest to call. Kurita would take the opposite of Davion in just about anything, but even Minoru Kurita would balk at facing even a depleted Regular Army and Davion's legions. Yet, honor being what it is and emotional pride built on honor would probably mean Kurita would fight the Star League cause, seeing this moment of disorder as the last opportunity to grab more power and larger share. Kurita wants Star League death. Pretty simple to see. He probably would know he couldn't win on his own, so would need help....
B. House Liao: Which brings up the Liao question. Liao is already tusseling with Davion (the Border War/Redfield campaign in 2760-62) would mean they were just itching for a final showdown. Barbara Liao being no fool would see the same thing Kurita would; this is the last opportunity to rid her House of the meddling Davions once and for all. However, being no fool, Liao is going to weigh things with a tad less emotion than Kurita. The Regular Army would mean the Davions would have help and the fight would not be even because even if Kurita was available to ally with, the majority of Kuritan forces are clear across the Sphere with only the tenuous Terran corridor as cooperative space. Would Liao ally with Kurita or would neutrality hold the day? Liao certainly would not ally with Davion in the cause of a Star League, but also may not stand in the way because Liao is likely to take it on the chin for doing so. If Liao joins with the anti-League forces, it does so somewhat unwillingly and probably looking to join the party as late as possible.
C. House Marik: Kenyon Marik does not like Kerensky. Kenyon Marik has distinct personality flaw in allowing his likes/dislikes to weigh his opinions, or so it would seem from the House Marik book. Kenyon Marik would probably decide on an emotional basis.
In politics, that is a no no.
Marik would be for a war on the side of the Anti-League. Parliment would be on the usual side - their own. The people would be Pro-League.
The actions of Davion would mean Resolution 288 goes no where in Parliment and Marik does not get discretionary powers because there would be no war looming to force Parliment to make that move.
Kerensky still being around means the Free Worlds populace, generally, is not Anti-League. The puts Kenyon Marik in a poor position to help Kurita.
If Marik wanted to help Kurita, he does so without the support of the majority of his state unless something serious were to happen...like an invasion by Steiner....
D. House Steiner: Previous to the Amaris Crisis, Steiner had a pitching economy (39 House Steiner PDF) and made Michael Steiner an unpopular guy during the Kerensky Regency previous to the Amaris Crisis because Mike wanted more mechs. Michael Steiner wrote in his memoirs he wondered about Council edicts and about the affect on the Star League itself. Generally, Steiner was held as not being very supportive of the League in the standard history.
When Michael died and Robert ascended the throne, there seemed to be degree of indecision. Robert is quoted as saying, paraphrased, he did not want to support the Star League versus Amaris because "Who knew if we would gain anything by helping General Kerensky?" If they supported or didn't support Kerensky, it did not matter, because as Robert Steiner saw things, control was paramount and opting out of the war preserved Steiner armies to maintain that control.
That and the rest of his statements would mean Steiner is not likely to support the Pro-Leagueists under Davion. Nor, however is Steiner likely to support Kurita's Anti-Leagueists. He is an opportunist looking to preserve his state and maintain control.
However, Steiner's populists and Estates General were firmly in support of the Star League. In the standard history, loyalist support was so strong as to conduct riots and such when they heard the League was dissolved.
In the alternate history we are talking about, the Loyalist cause is probably enough to rip the Lyran Commonwealth to pieces as the Federation of Skye and the Tamar Pact remained as political bastions. Or, it is enough to force the Steiners into support of Kerensky/Pro League/Davion.
However, Jennifer Steiner would replace Robert on the eve of the standard history Succession Wars, and she seemed only too ready to go war. Doing so though, might mean the general destruction or peaceful dissolution of the Lyran Commonwealth, with parts supporting the Pro Leagueists and other parts opting out of the war.
Conclusion:
Overall, I see the Council being "reformed" under Kerensky with Davion's full support and certain portions of House Steiner at first, assuming the Steiners don't get convinced and are ousted. A civil war is not wholly removed from being a possiblity in House Steiner.Liao might send a representative if only to maintain an air of uncertainty until Barbara Liao decides way to go. Marik is probably not going to attend until he figures out if he has enough support to do something against the Pro-Leagueists. Marik is going to probably try to side with Kurita only to throw his state into civil war, but he has to know that his state will lose whatever position it has in doing so. No one joins anything if Davion is put in command, though, so the Council would form under strain.
Liao is a wildcard though. Davion versus Liao is an even playing field to me, despite writer contrivance. Liao is unlikely to move unless someone else shows up to support Kurita. Unless Liao joins, the full force lands on Kurita.
Kurita is bound by honor to keeping pursuing the insanity of trying to fight both the Regular Army, the Davions, and to a lesser degree, Lyran Loyalists showing up on the Rasalhague border. Rasalhague might even try to break away from Kurita while the larger show plays out.
However, Kurita doesn't have a chance, not unless Liao or Marik get into the act. And either of those seems unlikely, Liao because they know they could not take on Davion/Star League units and Marik because he could not drum up the support to get Res 288 passed.
If Kerensky decided to stay with Davion support, it seems likely Kurita would fall, Steiner would be neutral unless forced into action either way, Liao would hold off until pushed to support Kurita, and Marik would be bogged down in inter-Provincial politics until either a civil war ensues (which Liao or Steiner might capitalize on again) or Marik is ousted.
Of course, this brief examination does not take other factors or even the actions of others (like the Taurians who might want their land back from the Davions). Yet, it seems less likely that were Kerensky to stay, a Succession War history that made Battletech what it is/was would ensue. There would simply be no popular support unless the last pillar of the Star League unity were removed i.e. Kerensky.
Note, that this all assumes Davion is willing to play second fiddle to Kerensky re-taking position as Regent until some political solution is reached. Davion is also unlikely as a choice for Regent - the only support it would have would be from House Davion with foreseeable results. That the statement comes out of the House Davion book makes me think that it was inserted as a "factional motivation" rather than a serious action by Cameron. People across the states would support Kerensky but supporting Kerensky does not mean they support Davion. But that is a whole other subject....