First a nitpick he did agree to being a constitutional monarch,
as for the actual question I doubt anything would have changed much other than they would have had to come up with a different reason to "imprison" him; it probably would have delayed his execution, but not for too long.
They were already enough radicals in power by then, they were just looking for an excuse and would have just found another reason, it may not have been as easy but I'm sure they would try to get rid of him soon. What might happen is the the radicals try to execute him, but end up discrediting themselves to many in the process, but I think this would just make things worse.Yes, but he tried to flee afterwards.
Couldn't the radicalism of the French Revolution be prevented had the situation not escalated with Louis's escape attempt, though?