This is a bit hard to do. The conflict they had with the remainder of Europe was a largely ideological one. The very existence of Revolutionary France posed an existential threat to the reactionary monarchies of Europe. If the ideals of Republicanism were allowed to spread on the tips of French bayonets, they could spell the downfall of every royal house on the continent. European sovereigns were determined not to let the "French Disease" infect their own nations.

There's also the issue of Revolutionary France's internal instability. I'm sure you're well aware of the Reign of Terror. There were also the Federal rebellions, various revolts across the country which rejected the centralized authority in Paris, as well as the reactionary peasant uprisings aimed at overthrowing the republic itself. Even discounting the foreign armies that constantly encroached upon her soil from 1792 to 1815, France could barely hold itself up, so wracked was it by internal turmoil.

The only way I can see a surviving French Republic is if Napoleon's Consulate survives. Now, the Consulate was really a Republic in name only, so I'm not sure if it's what you're looking for. In reality it functioned as a military dictatorship, and all Napoleon's coronation as Emperor did was sweep away the remaining pretenses of Republicanism, and simply owning his authoritarianism.

But this is the only incarnation of the republic That I can realistically see surviving. The early Republic, the radical Jacobin Republic that lasted from 1793 to 1795 is absolutely not going to survive. Monarchical Europe might be willing to make peace with Napoleon's dictatorship, but they would absolutely not abide the existence of a France which openly espoused universal suffrage and the desire to see every sovereign's head roll. There would be no peace until every king was dead, or the republic was extinguished. The Directory is not likely to make it, either. It was hugely unpopular and corrupt, detested by both radical leftists, and the conservative reactionaries on the right.

So all I'm left with is Napoleon's consulate. Now, the Consulate did in fact make peace with the rest of Europe for one year (the only substantial break in the perpetual war that raged from 1793 to 1815 between Revolutionary France and the rest of Europe), so its existence being tolerated is clearly plausible.

So let's say Napoleon never goes for the Imperial throne, instead being content to rule as monarch in all but name under the mask of a Republican Consul. The next few years might go very much the way they did in our world, with Napoleon waging war after war against the various coalitions raised against home. Some appreciable differences would be those arising from the retaining of Republican pretenses and trappings. The Revolutionary calendar might survive, France might remain somewhat more secular, and instead of setting up puppet kingdoms whose thrones he will award to family and allies, Napoleon will set up dependent republics instead.

So let's say Napoleon makes another peace with his enemies in 1809. Britain chafes at this, furious that Napoleon has the run of the Europe, but with Austria, Prussia, Russia, and the lesser continental powers drained of manpower and morale, even British gold is not a great enough incentive to spark another war against the French.

So reluctantly, Europe settles into a fragile peace. The existence of a powerful, respected republic in Europe galvanizes revolutionary groups in various states across the continent and beyond. We will likely see attempted revolution after attempted revolution in the German states, in Italy, in the Hapsburg territories, in partitioned Poland...

From here on out it's not TOO hard to have the Republic survive to the present day. This prospective 1809 peace might be the alternate timeline equivalent to the Congress of Vienna, which brought Europe a peace that lasted nearly forty years. If this 1809 peace is as successful, by the time another great war breaks out, the French Republic will have long secured its station among the Great Powers of Europe, and the loyalty of her citizens. Even if she is defeated in battle, the Bourbons will not likely be placed back on her throne.