I think what is crucial is to prevent the National Convention from establishing itself as the executive power in France in addition to being the legislature. IMO the best way to do this (after executing Louis XVI and creating a republic, but before falling down the slippery slope of radical revolution) would be to strengthen the Conseil exécutif (Executive Council). And to accomplish this, one would need to diminish the influence of the sans-culottes, particularly Jacques Hébert.
An idea occurred to me, just now. The POD is that after the Tenth of August incident, Jean-Marie Roland succeeds in moving the Girondin government out of Paris, to Blois - away from the influence of the sans-culottes and the revolutionary Paris Commune. Louis XVI still remains in captivity, and is still executed similarly to OTL. The Girondist National Legislative Assembly (NLA) in Blois is shocked by this, and as a result shifts slightly away from its more radical sentiments (short of royalism, but not as completely radical as the sans-culottes).
The abolition of the monarchy and the ensuing collapse of Paris into anarchy prompts the NLA to organize a new Garde nationale to strengthen their position. As Paris destroys itself in the Terror, the heavily Girondist National Guard marches into Paris, calling for the heads of Robespierre, Danton, Hébert, and Roux. The Communard militias are called up against the invaders, but they are divided and in some cases support Roland's government. After weeks of fighting, by late 1793 the city is back in Girondist hands. The National Convention is destroyed and the Committee of Public Safety is disbanded. The surviving sans-culottes are massacred or imprisoned, the Constitution of 1793 is thrown out, and so forth. The NLA moves to restore the power of the Conseil exécutif, with the express intent of electing the first President of the French...
How does that sound? This could stabilize the French government and allow the republic to survive. It's just something I threw together over the past hour, and IMO it sounds pretty cool.