He may but this promotion would not necessarily be meaningful for a long time. The problem with the loyal troops is that, with the few exceptions, they were foreign regiments on the French service. An overwhelming majority of the French national regiments sides with "the people".
Following alleged Nappy's advice of dispersing the mob (actually, not yet very "revolutionary") with a grapeshot would work as far as dispersing the mob was involved but it would also produce a terrible political reaction that would only accelerate the revolutionary process.
Not to mention that this action would not promote him (the 2nd lieutenant) into an army commander and even would not make him a general: in the Old Regime person's pedigree (and service record) mattered too much for an obscure noble from Corsica to make such a fast career just thanks to an action which would not be uniformly applauded even in the royalist camp.
You don’t need majority support to succeed in restoring order. Just loyal troops at the right point. You strike here. The volatile groups that make the more or less radical revolutionaries are going to be divided about what to do. You behead the radicals and most of the sheeple will scatter. Then you spin the event in the right way and 75% of the population will then support the legitimate king against the parisian mob.
That’s what Napoleon did a few years later. People wanted order before anything. And most peasants, that made up some 75% of the French population and the catholics and the nobility and the priests will rally around the king’s flag.
There was no support for a republican regime then. It is rarely known that, in the first legislative elections organized by the revolutionaries after they abolished the monarchy by a violent coup, the turnout was merely 10%. They would never have won a majority to support their radical methods and program if the election had been free and if the non-radicals had not been terrified or disinterested or disgusted by the chaos in which France had begun to sink.