I avoid public bathrooms, but when I do use them this is what I do.
1. pee (I use a pstyle(i had to cut it smaller) and yes the travelmate is longer)
2. listen/ look(if at urinal) for other guys in the room
-if no one is there, then clean it as much as I want
-if someone is right there at the sink I will either...
a. just put it in my cargo pants pocket, pee is sterile anyway
b. put it vertically in the palm of my hand and keep it from view as I go to the sink, and rinse it while washing my hands and slip it into my pocket like its no big deal.
but really, what guy is gonna break the guy code and talk to you in the bathroom? even if it looks like you are washing something, and remember, sometimes guys do need to use the sinks to wash off something.
I washed out an stp with soap in a busy airport once, no one even noticed, they were too busy doing their own thing
I think trans guys luck out with the bathroom situation, because in womens bathrooms, the girls often go there to socialize and they actually look at, and often judge the other girls in there