If your inspection protocol calls for a visual inspection, there's a reasonable chance they'll notice that it's missing. It's almost impossible to predict whether you'll pass the sniffer or not. Testing procedures vary even from county to county within the same state, let alone from state to state. Kind of humorous that someone from Massachusetts is telling you you'll pass in TX -- the tests, equipment, training of techs who perform the tests are different everywhere you go; not to mention each vehicle is unique. Passing in one area is no guarantee that someone else will pass elsewhere. I've seen posts from people with stock cars in good working order with all smog equipment present that failed, and then you've folks like those above that indicate they've passed with emissions equipment missing. Best thing to do is to find out exactly what they techs that do the inspection are looking for before you take it in for the test. If they do a thorough visual inspection and the pump's not there - well, you can figure that out.
The pump is designed to supply excess oxygen to the exhaust stream so the cats can 'catalyze' the chemical reactions needed to turn unburned hydrocarbons, NOx and CO into non-controlled emissions. Certainly if you don't have cats there's no reason to run the air pump. And the cats won't work as well if you have them, but you're missing the air pump - they're designed to work together.