My sleeping pad is a NeoAir, but the volume is similar to a BA pad. It takes 15 deep breaths to inflate to where I usually want it. That is maybe 45 seconds. At the volume of a frame pump it would probably take thousands of strokes on my bike pump.

I am guessing that maybe the OP has a pad with down or some other insulation that he wants to keep dry, hence the reason for not wanting moist breath in it. Still a low volume pump like a bike tire pump sounds like a very poor answer.