Only problem is Lafayette had not the will nor the capacities for becoming a ruler, even less a dictator. Even if he took the power, his political aim was a constitutional monarchy and it is not possible to put this into order with Louis XVI as king.
I disagree that a Consti. Mon. was not possible with Louis XVI. At the end of the day, the King wanted to keep his head and what remained of the trappings of his presiptigious family. If you had a powerful but still reasonably authority figure like a dictator Lafayette you could keep the king in line (whether that would happen is another matter)
If he still wanted constitutional monarchy - Lafayette could offer to act as regent for Louis-Charles, the Dauphin, until he becomes into the majority. That is IF the dauphin would be acceptable to those who wanted to keep some vestiges of the ancien regeme, if the little brat is still alive at the time, and if Lafayette could hold power as temporary dictator. Of course, there would be other butterflies if Lafayette becomes dictator (even if temporary one) - Napoleon might not have much a military carrier, unless he becomes quite ambitious and want the title of dictator for himself.
The idea of Lafayette as Regent-dictator to a child-king Louis XVII is a good one, as it gives Lafayette the political coverage and authority while also giving him a decade to sort out new French institutions. This may be my favourite way of having Lafayette as dictator in the revolutionary period.
If he’s another sort of dictator, then the stability of a post revolutionary regime is in gaining logevity - the longer a system is in place the more entrenched it becomes - and through operating under a reasonable and understandable set of principles and rules.
The first you might achieve with a new constitution - likely best adapted from the 1791 one. Lafayette’s position need not be included in it but it must be clear that he was going to be de facto in charge for the time being. You don’t want a dictatorial precedence if you can avoid it (even while having a dictator)
Alternatively you need the absolute loyalty of the biggest guns. To which Dictator Lafayette would need to quickly co-opt the French Army to retain power. The National Guard were by and large too undisciplined to be the foundation of a enduring dictatorship. You need the guns to restore & maintain order and to allow for the National Assembly to finish its job of dismantling/replacing the ancien regime’s governing structures.
The pros of Lafayette as leader of France are that he is a both a noble and a Republican, as well as a pragmatist and a war hero. At least before the Champs de Mars, he’s an acceptable choice for all major factions except the most radical of the revolutionaries. The cons are that he is too principled to be a populist but too much of a populist to remain consistently principled. He love to have the love of the people but when it was unclear what action would get him that love he was indecisive to the point of ineptitude.
OTL Marie-Antoinette said of him she'd rather be dead than owing him her freedom, to state the consideration he enjoyed in the royal family...
I bet that’s just because she’s still a snob from when Lafayette and Marie Antoinette danced as teenagers at the royal court, that the Marquis was such a bad dancer that Marie Anotinette would laugh and mock him. The whole court back then thought Lafayette a bit of a country rube for such uncourtly ideas like loving your wife.