I feel like there is a lot of over-thinking going on here, which is understandable, but it's taking on a negative tone.

The cargo pants/lots of tissue sounds like a good remedy. Or, like someone said on a past page, you could avpoid going out to public areas that would involve you using your preferred restroom during your period. This, however, can be difficult unless you are lucky enough to have a short period. I know I don't, and I know my bathroom life would be a lot easier if I did.

Just do what makes you comfortable, y'know? Go into the mens stall when it's kinda late or no one is there, feel it out, figure out a plan of action before you end up having to do that AND not give yourself away.
And one last thing. Not all women give off an odor. I've lived with women all my life, and most of my friends are women, and I can't honestly say I've "smelled" them. Perhaps a stranger in women's bathroom yes, like ONCE OR TWICE in my entire life, but I doubt it is something to fret over as long as you do your best to stay fresh, which most tampons and pads are getting better at doing.