Constitutional monarchy seems like the most likely outcome, assuming Louis XVI and the sans-culottes can both be convinced to go along with it. The sans-culottes can probably be won over by the classic combination of bread and circuses, while the main thing Louis XVI will need is a reasonable reassurance that Constitutional Monarchy isn't the first step along the path to a Republic.
The biggest issue Lafayette is likely to face is that he has a fairly narrow window of time to work with. If there's going to be a functional Constitutional Monarchy, it needs to happen before the Flight to Varennes and ideally before the Day of Daggers. However, until around the time of the Civil Constitution of the Clergy there were not really organized factions within the pro-Revolution groups (including Lafayette's Society of 1789). That doesn't give him much time to take power and seriously shift most of the major players in the Revolution before things move past the breaking point.
Lafayette's best chance would probably be to work out an alliance with Mirabeau. They couldn't quite work out an agreement in OTL, but if they manage to come up with something then that would give Lafayette a political genius to help him out, and assuming his death isn't substantially butterflied Mirabeau should conveniently die right around the time he and Lafayette finish consolidating their power, which would leave Lafayette in sole command of the Revolution.