Bettas are anabantoids, they have a labyrinth organ, much like gouramis. They don't need as much dissolved oxygen in the water as other fish, because they can breathe right from the surface. Obviously, they still use their gills, how else would they rest? Plus, the oxygen obtained from the surface isn't enough.
to make a long story short, a bubbler is not necessary, and would probably just tick your betta off. They like as still of water as possible. Do not confuse having surface area with not having lids. Bettas are just fine in tanks, and actually NEED lids, as they jump. Just so long as there are some holes for air circulation. Just like if you bagged any kind of fish with all water and no air, it would die, same goes with a betta.
If you bagged a non-anabantoid fish up with no air, and a bubbler, they would be ok, but bettas rely on breathing from the surface, and a bubbler alone would not be ideal. Catch my drift?