To be honest, I've wondered about the mechanics of air getting into the bottle too.
If the sprayer sprays on the very first pump ( which is what happens most times I spray) that would mean that "replacement air" is not going back down the sprayer tube, or like the OP said, you'd find yourself doing a bunch of non-productive pumps each time, to get past the air that was hanging out in the tube.
Does that mean that somehow air gets in via some other entry way, other than coming in through the spray nozzle ?
Whatever the case, common sense would indicate that in totality, no more air will be in there than what a person actually sees in the bottle. I suspect that could mean that in bottles that were close to empty, there would be more air for the remaining juice to react with than in fuller bottles. But my assumption is that the air that is in there has already been well-saturated with gas from the perfume for quite a while, and is probably pretty inert.
No matter how it gets in there in the first place.