A few things I'd take a look at:

Is the garage at a higher temperature than the outside air; by how much? Are the garage and its attic well ventilated? Is the garage open to its attic?

What color are the garage doors? White reflects heat best.

Are the floor and wall of the bedroom well insulated?

Can use a whole house fan

I am trying to figure out a way to do the same, only with a 4-bay beneath a flat roof. Saw this in a mailer from a well known warehouse store...seems reasonably priced. Don't know if it will work for you, but something you may want to consider.

While an through-the-wall exhaust fan could certainly be used, the air it removes needs to come from somewhere for it to be effective. That means leaving the garage door open slightly or adding some kind of opening somewhere like a window.