It wouldn't be first to Paris wins. The stability of Europe depends on no large scale war breaking out which would've happened if someone had tried to impose their choice without negotiations. Because of this, the powers of Europe largely agreed that a restorationist approach was best for the stability of Europe and the survival of their own thrones.
Even if you can feasibly kill all the descendents of Louis XIII, the next legitimate inheritors as far as I can tell would be the House of Bourbon-Condé with the heir being Louis Joseph, Prince of Condé and prince du sang of France as a descendent of Charles of Vendôme grandfather of Henry IV. After the House of Condé you'd also have the House of Conti.
The thing about French Salic Law is that it basically prevented a foreign dynasty from gaining a superior claim to the throne than that of a native French dynasty regardless of how far back you had to trace the line. There will almost always be a Bourbon cadet branch to draw upon and the French knew it which is why they had the rank of prince du sang to identify possible back-ups monarchs.
But let's say screw it, the Bourbons couldn't be trusted to hold onto their throne so we need a non-French dynasty to rule, and if you want legitimacy just marry the agreed claimant to Marie-Thérese.
If you start tracing the French monarch through female descendents of Henry IV (having killed off the descendents of Louis XIII), then the Spanish Habsburgs if as you said are still extant would have a decent claim. But, the British and Prussians wouldn't tolerate a Habsburg on the throne of France.
After the Habsburgs would come a Savoy claim but I doubt if there would be much support for Savoy to grow massively in power overnight amongst the other European powers (you can't win a war against France and have the French leave with more territory, which'd be the case even if you ensured that the same person didn't sit on both thrones).
After the Savoyards comes my favourite option: the House of Stuart. The only surviving claimant would be Henry Benedict Stuart, Cardinal-Duke of York who was 68 in 1793, which I think would be a hilarious outcome if he became Cardinal-King but this would only postpone the succession crisis and anyway the British would not countenance a Stuart on the throne of France.
Oh, right, there can't be ruling Queen of France. I totally forgot about that. So that's why I looked at the first relative with prestigious ancestry.
Louis Joseph Prince of Conde and Prince du Sange seems like a perfectly good choice. I thought he was descended from Louis XIII? I can't remember. Ok, let's say he's not. So the Hapsburgs and the Hannovers agree he is the rightful claimant. Since they dispersed the conscript armies, the powers to be crown Louis Joseph as Louis XVII of France and Maria Caterina Brignole as Queen Consort. The old royal family, in laws like Marie Anotnette (now widowed) and daughters Marie Thérese of France stay as esteemed guests or something, Louis Joseph must have felt sorry for what his relatives had to do.
I'm quite sure the captors who killed Louis XVI but decided to spare the rest when the Austrians got close in hopes for pardons are... not going to get pardons.
And then someone needs to pay the bill for all the destruction the Austrians caused in the cities when they blamed the city folk for the revolution. Who's paying for Marsielle, Toulon, Hyres, and Lyon after the Austrians took anything portable and edible? If the crown pays for it, they would need to raise taxes which... well that's just asking for a repeat. Let's say the royal treasury does not get involved. I guess the locals are going to rebuild with their own money. Maybe they can beg from the peasants that Austria had mercy on?
First rates are expensive ships, I think 3 times as much as a third rate... before you crew it and put guns on. Maybe they can sell some of them to raise funds.
If Louis Joseph Prince of Conde is not a descendent of Louis XIII, I guess that's all my thoughts for this post. The rest doesn't mean much.
If Louis Joseph IS descended from Louis XIII, then let's say he died in the revolution too.
Now what? Well, we agreed that a restorations approach will be done, so the big players will find a claimant.
I needed a way to get Charles IV of Spain out of the way. If he died without heirs, that line isn't usable for the French throne on the account of being dead. If that line didn't keep the throne because the Hapsburgs won the succession war, then that actually means the Austrian Hapsburgs would be sitting there, as the Spanish line went extinct, that's what caused the war to begin with! I don't think the Austrian line has a claim to the throne of France. If they do, I think the British might be OK with it. Hannover and Austria always got along well (they were on the opposite sides of the Seven Years War, but they were never at war with each other and the Hannovers even sold supplies to the Hapsburgs, although that's not exactly a gift) and the British Monarch are Hannoverans. The Hapsburgs felt betrayed at the War of Austrian Succession when Britain said "look, we fought for most of your inheritance, we can't fight for Silesia" but there wasn't anonymity from the other side. If the Austrian Hapsburgs do have a legit claim (I think only the Spanish Hapsburgs do), the British might be OK with that as long as the Hapsburgs continue to make friendly overtones. France was always a rival, having France be used to serve Austrian continental interests instead of opposing British commercial and colonization interests would be nice. On the other hand, if the new Hapsburg King doesn't redirect France's overseas expansion funds towards rebuilding France and severing Austrian continental interests, the British would regret letting that happen. So maybe they won't let that happen just in case the new king does that!
OK next comes the Savoy claim. Actually, the Hapsburgs and the Hannoverans considered the Savoy friends. I think they wouldn't mind them becoming a power. Maria Theresa from a generation ago called them honest. polite, and friendly. There were intermarriages. The only problem with this is that France lost a war, why should it gain land? Savoy is a Duchy and someone who has that title and the Kingdom of France ill consider himself the French King first. As you said you can't win a war against France and have the French leave with more territory! Maybe split it between different dynasts?
And most amusing the Stuart claim. Is there anything wrong with an unmarried Stuart being King of France? He's not going to start a dynast there as King-Cardinal. That would be absolutely hilarious?
OK, if Louis Joseph is descended from Louis XIII, then what claimant do you think the British (Hannoveran) and Austrians (Hapsburgs) will agree on?