I mean, one way Robespierre can spin this is that the legitimate government has managed to overcome the military coup d'etat that every political figure, including him, feared would happen if there was a war. Let's not see this as a black/white conflict, what Brian and Napoleon did was illegal in every sense of the term.
While correct, there is also the fact that every one of our political allies and our family (Biological and adoptive) is on the shit list of Robespierre, and if he could have it his way without causing a great deal of backlash and cause the entire army to just straight-up end the Republic, he'd kill them all.
Some of them, like Brian and Nappy for Legitimate Reasons and Treason Charges. Louis and Charlotte? To deny the royalists easy access to two of the most legitimate royals that are still in the country, that he can get to. Especially if he executes Louis while he's still excommunicated, and plays that up.
Others like Kellerman, and even Davout? For literally nothing more than their connections to the old nobility in any way.
Ney is a special case since everything he did was legitimate and what the Directory did was a crime against the army, but he'd imprison him so that he can give him a deal, and calm the army down.
He'd also make us resign our commission because we literally are the most popular general in France, and frankly, our history of disobeying and bending the orders we are given makes us seem politically unreliable for his government, regardless of the results we produce that are strategically and tactically valuable for the French State.
It's a nice bit of legend, but Napoleon managing to have the army follow him wasn't because he had 11/10 charisma, though he was plenty convincing. It was mostly because the army was already full of Bonapartist supporters and the Bourbons were doing a hell lot of mistakes in actually keeping the army loyal.
The fact that Almost every big-named officer was a marshal that he created also helped.
I really think we shouldn't exile Robespierre. The fact that he managed to worm his way out of any blame for his actions during his first rulership at the trial is proof enough to me that he's got enough Charisma to make use of the exile as another chance to recover power. Furtheremore, the enemies of France can use him against us.
In terms of our political situation from other nations:
Robespierre taking power is literally the one thing in the British foreign office needed to more effectively isolate France from the rest of Europe.
His reputation, both for good and ill, as a radical and a fervent believer in the Revolution, will not make other heads of state happy, nor would they want to deal with France. and the Brits can use that to their advantage.
When it comes to handling Maximilien Robespierre in a non-peaceful way, I am of the opinion that we either jail him somewhere where he is unable to escape or we execute him. France needs stability more than ever and Robespierre has proven himself to be a disruptive element, therefore he must be removed in a way that prevents him from returning to power ever again.
The thing about Robe right now gaining power, is that it was mostly legitimate.
Brian resigned and Robe took command before Nappy could seize power, in what could be considered a bloodless coup.
Robe is a revolutionary yes, but he's one of the smartest politicians of his era, even in some respects, better than Nappy.
I'm pretty sure the kids aren't gonna get executed, I mean, no one at the time actually entertained the idea, plus, they are very useful as hostages. Its more likely they'll be put under intense surveillance than anything.
I mean, you say that now, but Charlotte is a legal adult in France, and Louis is now a hardened killer, who's so willing to die for France that he wanted to kill himself to do it.