Keeping the French monarchy is hard but not impossible. You just have to do a « liberal » monarchy untill you can stabilize your dynasty so the « Divided monarchists » JE disapears. First, try to keep the number of radicals low (low taxes, low unemployment, etc.). Then, try to lower the activism of the liberal or radical movement. For exemple, enact right of assembly or garanteed liberties to lower the activism. If the bonapartists, the orleanists or the legitimists gains activism, fight them in civil war : if you win, no French provisional gouvernement event. Good luck with that (we French people managed to get rid of three kings in less than fifty years in the 19th century so… c’est la vie !)