I personally have gotten to the point where I really just don't care about what someone else may or may not think. I keep my spare diapers in my car, and take them -in hand- to the restroom. Amazingly I have yet to even get a second look, let alone a stare/comment.
Bathrooms are the one place where noise bounces around and echos a LOT. The tapes, and even the crinkling of a diaper will be obvious to anyone else in there. Unfortunately you are going to have to adapt to not caring, or complete paranoia. It seems in a public bathroom there is no middle ground. Luckily, most everyone else around you already doesn't care.