Philippe Égalite hasn't voted for Louis XVI execution (how could Legitimist support claims of descendants of king's murderer ?)
To be honest, even if Orléans wouldn't have pulled that (it arguably disgusted even Robespierre), he would still have seen as one if the instigators of the French Revolution, either trough interested manipulation, or at least sheer blindness.
It would do its part, but wouldn't make bad blood and most of all, huge ideological differences disappear at the latest.
There is short period of republic after deposition of Charles X. So Louis Philippe is not blamed as much as IOTL for fall of the main Bourbon line by Legitimists.
That's an actually intuitive idea there : I like it.
It would strengthen republican ideals (a bit like the short-lived IInd Republic did IOTL) in the same time it would make French monarchism more unified (and maybe *Legitimists more porous to some liberalism), but you'd have at least a more credible monarchical alternative by the end of XIXth at the leastest.
Other option for compromise is extinction of both lines and survival of either Condé or Conti line from which candidate acceptable for both sides could be chosen.
Thing is, you see, the dispute was far from being only dynastical : one could even say it was more how it looked like and less what it was.
Legitimism is above all a dynastical loyalty, and while it make it porous to various influence in the name of dynastical legacies (a bit like Carlism in Spain), will focuses more on anti-liberal politics in the name of either traditionalism, either direct link between a paternalist king and his people above politics (meaning a certain anti-parlementarism and anti-partisan stance).
Orleanism, however, is more "political". Orleanist representent are supporting Orléans less because of fidelity to their line or because of an immanent virtue of its chiefs, than because Orléans are favourable and readty to support liberal, parlementarians ideals.
We have two different conceptions of, not only monarchy, but state and politics there.
Giving the "ultra" nature of Condé and Conti in the early XIXth, you'd have basically Charles X on steroids; which is going to frighten a lot of deputies, politicians and journalists that were IOTL Orleanists; while Legitimists (even liberal ones) would accept the fact in the name of dynastical legitimacy.
ITTL, then, most of Orleanists supporters may look at another house but no one would have the prestige of Orleans and their reputation of liberals, and would likely be so far of the main succession body that it would be more or less pointless to call it a dynastical succession. In the end, most may simply abide by a liberal Republic (and powers as Britain may even think it would be better than a Carlist-like France).