"Assuming it doesn't draw more power" has got to be the problem here, right? I don't know much about wireless technology but from a purely physical stabdpoint, faster signals means higher frequencies, which means higher energies, which means more draw from the battery. Yes, shorter active time means less draw, but it's like that swiss cheese joke:

Swiss cheese has holes.

More cheese = more holes

More holes = less cheese

Therefore,

More cheese = less cheese.