If your friend believes in Newton's laws of motion and it is a logical person then he is doomed. Flat Earthers sadly are completely illogical in their aproach to reasoning.
Your argument is not correct and is a violation of Newton's 3rd law in fact. If Brownian motion of the gas particles is responsible of excerting the force that uplifts something the reaction force is not on the object but on the gas particles. You said that the gas acts on the object and then a reaction force happens also in the object that counteracts this influence but by the same argument there shouldn't be any kind of motion in the universe: if a push a box a reaction should appear that opposes my push. The thing is that Newton's third law establishes that the reacting force is acting in the object which excerts the action. Is not the same object. So if we assume brownian motion could be responsible for that then your friend is correct, a change in motion is expected.
The problem is that brownian motion can't excert any consistent push by definition. It is a random process of interactions. The overall motion would be zero since the same amount of gas particles is pushing the object in all directions on average.
Flat Earther's are also mistaken on the idea of bouyancy replacing gravity. Bouyancy IS itself gravity! An helium balloon "floats" because gravity is pulling both the ballon and the atmosphere to the ground but in doing so it is pulling the air with much more force than the balloon since it has more mass per cubic centimeter (which not only corroborates the existence of gravity as a phenomena but also the theory of gravity developed by Newton which states that the force is proportional to the mass of the object). Since the air is pulled to the ground more violently and since air is a gas and can fill any gaps it is occupying the volume where the balloon would be if there was no air at all: the ground.
This stupid idea that bouyancy is explained by relative density instead of gravity shows the lack of doubt in flat earthers. They don't question the idea of less dense objects going upwards and denser ones going downwards, they don't understand why this happens, they just assume that is like that (in general it happens) without any explanation because it was said by some authority (maybe at school). They don't understand that actual laws make that behaviour a consequence, instead they think of the consequence as a law.
The same stupid thing happens to them when they talk about sea level. "Water levels out". This is true in a wide variety of context for sure (just as the fact that denser things go downwards) but it is not "the law" it is just a consequence of applying the law in a specific context. In fact the law (which in this case is the law of gravity again) can go against this idea of water leveling in another context. Water levels because the gravitational field is uniform, but at large scales you come to realize that gravity is pulling to the center of earth and thus it is non-uniform (the direction of the force is different in different places relative to other), Water leveling is just a specific situation, a consequence of these laws, not the actual law you have to belief without any kind of skepticism.