I tend to think that any event with as deep and lasting consequences as the Revolution of 1789 has deep and persistent roots. Sure, rebellions of various kinds flare up all the time--but when one topples an ancient monarchy essentially permanently (it recurred, but that didn't last) and then goes on to unleash a continent-shaking series of consequences, it means that there was fuel accumulating for the flame. It's actually a lot like brushfires in Southern California--sure you can stop a fire that is just starting to get out of control right away. But the dry brush just accumulates then still further and eventually there is an even bigger fire. It is the fires themselves that hold their overall, average intensity in check. Or like earthquakes--the pressures are always building up, more or less steadily, then get released quite unsteadily--the longer you delay the Big One, the worse it is when it hits.
One approach then is actually have a revolution of sorts flare up earlier, and then its consequences cascade less spectacularly. The Bourbon dynasty might be ended but some other dynasty quite similar to it, but smarter or working in different conditions, emerges and lasts longer, perhaps even transitioning into a more modern mode of operating gradually and systematically rather than as the aftermath of a huge sudden social revolution.
Another is to address the dry brush as it were, to ask what the Bourbons could have done to bring their society back from the brink--a brink they didn't even realize they were stumbling near, that makes it harder. And it wasn't just the ruling dynasty of course. Louis XV, so I've recently read in some other timeline, was actually concerned to try and shift some of the burdens of the French state from the lowest classes to those more able to pay, who were also beneficiaries of the system--to the nobility in fact. But they frustrated him at every turn. WI frustrated with them, he had led a sort of revolution from above, mobilizing the lower classes to break the overweening privilege of the nobility and impose a more British sort of social order in its place? He'd be playing with fire in a big way to try that, he might wind up being a victim of his own deluge instead of just letting it build up "Apres moi!" as OTL.
But of course the great Revolution did not just transform France--all of Europe was headed for a massive social earthquake driven by accumulating deep shifts in the basic organization of society all across the board. Louis XV might ride his tiger to a quasi-Napoleonic sort of hegemony, under the Fleur-de-lis instead of the Tricolor; perhaps happening earlier he wouldn't get quite as far but by that same token the conquests might be more durable? A less radical version of the Code Napoleon, perhaps a hybrid with a lot more Ancien Regime carryovers, imposed on a somewhat smaller segment of Europe but that segment remaining under the French crown despite drawing a desperate alliance of the other European powers against him? Maybe Spain goes along more willingly and stays in the reformed Bourbon coalition?
This is yet another variation on doing it earlier and getting it over with sooner of course. I really think that somehow putting it off another decade or generation just means that the fire is even more devastating when it does break out, and maybe it starts somewhere else, but I do believe Europe was necessarily heading for some kind of vast social earthquake.
If the goal is to avoid that, you'll need Uber-Bismarck, Super-Stolypin, Pitt the Greatest, superhuman ministers of the Ancien Regime all across Europe, working carefully with each other to either arrest the advance of capitalist transformation (thus bringing Europe into an age of repressed stagnation--Maetternich the Maerciless!) or manage the transition with incredible skill, largely to preserve the mere names of old institutions and perhaps some quaint colorful customs but midwifing the remarkably painless birth of a substantially new Europe.
That is one hell of a long shot, and the more realistically done it is, the more you'd just substitute persistent ugliness of the Old Regimes carried over for the sudden revolutionary ugliness of a violent age that did however sweep away a lot of old trash. To get everything good and nothing bad, you need Space Unicorns of surpassing wisdom and grace that all humanity bows down to and adores.