I hope to ask the escort if a rapid screening for infectious diseases can be conducted after entering the room (using a test kit, with results available in 15 minutes), but such a question might be considered offensive. I wonder if I can do this?

Some might consent if you do the same. And some would definitely tell off. Are you planning a BB? 

No BB

If that's so important to you, just ask them, don't worry about an outcome. There are plenty fish in the sea. 

What kit is that I haven't seen it?

With an escort ask anything you want they are used to it.