Although the Paris Commune has gotten a rather notorious history due to the overall course of events, its position as a visible failure helping to guide future socialist revolutions, etc... the actual goals of the Paris Commune, particularly early in its formation, were relatively light. They wanted for Paris to be a single municipality, to maintain the National Guard salaries, to not have to pay the back rent incurred during the siege, and to not have the national government scurry off to Versailles, without any clear overall program of grandiose socialism or anything like that. While there were definitely radicals inside the commune, both neo-Jacobin, Blanquist, Proudhonist, and else, a big part of why it went so far to the left was that Versailles was completely unwilling to compromise on anything with them. The more conservative people of Paris generally didn't participate in the elections because Versailles stated they were illegitimate, and respectable republicans who did get elected ultimately refused to hold their seats. Even then, the council's majority was dominated by neo-Jacobins, not Blanquists or Proudhonists.
The major sticking point in any negotiated peace seems likely to be the National Guard. Although in March 1871 they are basically the only military force in France, thanks to the Prussians, they are also very left wing due to their nature as a paid militia for a city under siege which is now conducting election of officers and other such things. Even if the government is willing to submit itself to the Parisian mob (as it was for most of French history), "the Parisian mob but also it has all the guns in France and is quite left wing in a republic that just got a majority of royalists" is not exactly something anybody is going to sign up for.
So what if, with a little bit of difference on the historical events [1], you wind up with a negotiated settlement and no actual exchange of gunfire which is something like:
1. No one is liable for rents incurred in Paris over the course of the siege.
2. There will continue to be a salary for members of the National Guard, but the vast majority will be demobilized - they will get some kind of pension or temporary salary so they don't immediately become homeless, but there's not going to be 200,000 National Guardsmen in a city with a population of 2 million in a time of peace. The government may intend to retract the salary at some later date, but would prefer to wait until things calm down first, rather than follow the zero-to-eighty trajectory that lead to the Paris Commune's insurrection.
3. Paris can have a municipal government of its own.
4. The government will remain in a temporary capital (Versailles? Bordeaux? Tours?) for a while, then move back to Paris. The intent on the government's side being to make sure the Parisians keep to their end of the bargain on the National Guard front, and that the municipal government isn't overrun by crazed radicals who will hack off their heads given half a chance.
The municipal elections wind up electing a broad swath of the left, but being actually backed by the government, the (near?) majority winds up being the more respectable sort of republicans, followed by neo-Jacobins, and then the remainder stretching into the far left - Blanquists and Proudhonists and the rest - who are firmly in the minority. As such, the government does wind up coming back to Paris to set up shop later in 1871 (after it becomes clear they're not going to have a king to behead any time soon thanks to the intractability of the Count of Chambord).
Obviously this is kind of putting all the problems off into the future, but that's what interests me: what does a Third Republic where the Parisian mob and the National Guard are still real - and now quite left-leaning - political forces look like? It will also obviously change the way things go with the First International.
[1]: Probably want get rid of Thiers. The theft of the artillery which kicked off the lynching of Clement-Thomas and Lecomte, was pushed through by him, against the advice of everybody else, basically the night before it happened. Without that, maybe things can be calmed down.