It is a pity that this post has not received any response as the questions asked are interesting. So I allow myself to revive this thread, hoping that it will be seen by others.
At the moment I'm thinking about how Louis Philippe d'Orléans could have kept Henry V on the French throne, but reading the course of events, if Louis Philippe had had the desire to simply suggest that the young Dukes of Bordeaux could become king, then the reaction of the Parisians, who surrounded the Luxembourg Palace, on August 3, 1830, would have been negative enough to discredit him in their eyes. Also, the great liberal figures who had offered Louis Philippe the power retained a mistrust, in particular LaFayette, and the idea of a Bourbon succession under the aegis of the Duke of Orléans could be perceived as a means of diverting Charles X to regain control of the capital. Because at the same time, he is still in France and although he has abdicated, he still represents a threat to the Parisians. Thus, as Louis Philippe I will say, to defend himself from not having proclaimed "Henry V": "I would have been chased away with them". Logically, I see the republic being born, not on the first day of the revolution, 27 July, in the din of the barricades, but after a misstep by the savior of the monarchy, between 3-4 August.
Louis Philippe had then said that he would not use force if an insurrection were to break out again in Paris. If this can be a political maneuver aimed at reassuring the radical elements, I have it more as an honest word for the centrists because it would then be difficult for them to order the national guards to fire on the population, see brothers in arms, barely a few days after the start of the Revolution. Surely he will leave his post and flee to Dreux with his family, whatever the Duke of Chartres could, by his ardor and his liberal spirit decide to stay, see to side with the Republic or simply be captured. Consequently, a new government would be created at the same time as the Republic is proclaimed, surely in an attempt by the deputies to control the events by constituting themselves with the Chamber of Peers as the National Assembly and offering to the Marquis de LaFayette, the head of the government.
Similar to the Second Republic of 1848, the government will consist of the central figure of LaFayette, who will bring in "sincere Republicans" and "despite themselves Republicans" to his left; Godefroi Cavaignac, Dupont de l'Eure as well as other new names for the former and Adolphe Thiers, Odilon Barrot for the latter; alongside “conservative” liberals like De Broglie, Laffite, Sébastiani. The policy of LaFayette in the first weeks could then be to drive out the Bourbons (the family of Charles X), the purge of the administration and the judiciary in favor of at least liberal figures. The Republicans are surely pushing for symbolic but important reforms such as the abolition of slavery, the return of universal suffrage but also international proclamations of aid to European nations. Elections will surely be put in place and would give a majority to LaFayette's moderate Republican current.
Obviously very negative. If the Prussian and Russian embassies agreed to a dynastic change, the republic would obviously recall all the shit that the Revolutionary Wars caused. Surely the Holy Alliance will discuss an armed response to the advent of the Second Republic, but I think there will be a certain paralysis on their side, especially with the Polish insurrection of November 1830. Also, I think that LaFayette will seek above all to calm the fears of the European monarchies, in particular of England which is the most to be feared, especially in the inevitable Belgian revolution.
However, the Republicans will push for intervention with the aim of taking revenge on the world of the Congress of Vienna, by wishing to cover the "natural borders" of France. Nevertheless, LaFayette could hold but if the Dutch intervene as OTL, then he will be forced to intervene as OTL himself in an objective of internal stability, under the pressure of the nationalist republicans. Here we could then have a republican or simply non-existent Belgium, annexed by France or shared between powers. Talleyrand's partition plan could be advanced, although it is flawed and could be changed.
The army will surely rally to the Republic, the senior officers could be a little reluctant but the bulk of the non-commissioned officers as well as the troops are either former soldiers of Napoleon or young people while the most reactionary are either too few in number or quite simply offside (I am thinking here of Marshal Bourmont who was in Algeria at the time of the Revolution). The prospect of combat in Belgium completes, I think, cementing the army for the benefit of the Republic with, in addition, a head of state like LaFayette.
The population seems to me more complicated, I think that the province will rally passively to Paris, and surely the big cities, as it can leave the crorie at the very beginning of the revolution, will surely be all acquired by the Republic. Obviously the populations of the West will perceive this with a lot of suspicions but the rapid departure of the royal family can paralyze them and one can imagine that the Duchess of Berry tries a restoration and provokes like OTL the vendée war of 1832 but surely it will fail like OTL.
However, dissensions will appear in the Republican camps. Surely there will be apparent contradictions between radicals and moderates. September and October 1830 as well as February and March 1831 saw major demonstrations by the Parisian population and Republican clubs, which may constitute a place of confrontation between LaFayette and his left, where the marquis may decide to use the force to put an end to the agitations (I estimate that the funeral of the republican general Lamarque, 5-6-7 June 1831, can be the equivalent of June Days uprising of 1848). The revolt of the Canuts can also occur and be managed completely differently, by which I mean more brutally.
Then a constitution would surely be given before or after, in what form? No idea. LaFayette could be elected President of the Republic by direct or indirect suffrage but would die in office in 1834.
Aiglon can come into contact with certain French deputies or even with LaFayette himself, and end up really investing in French politics at the risk of causing a heart attack in Metternich and Francis I of Austria. Would he seek to go directly to France or to join his cousins and uncles in Switzerland? Maybe the Bonapartes wouldn't be banished from France. But I do not see here the princes taking power in 1834 but much later either in 1836 in a successful uprising in Strasbourg or in 41 in Boulogne. The Republic can then subsist but perish in the face of the political and social crises that cross it.
Conversely, the Duke of Chartres (I come back to my original idea) can also appear as a popular liberal monarchical firuge by serving the Republic but with a constant suspicion hovering over him. It's as vague as the establishment of a Second Empire.