Regarding the sports bra, that's obviously totally up to you.
The way I would look at it is, is there a real need to? If yes, then obviously wear it.
Otherwise, is there a valid point that makes sense to? Personally, I wouldn't, but that's just me. When I'm out in guy-mode like this, I don't really aim to call attention to myself. A sports bra would "scream" that person is wearing women's clothes. I know there's a decent chance I would get looks, whispers, fingers pointed, laughs, maybe even some kind of comment. To me, I don't have the need or desire, nor would I think it's worth it in my case.
When I do this, I typically just wear women's stuff that's kinda-sorta gender-neutral, but a bit on the femmy side. No bright "girly" colors or floral patterns or lace or whatever, no skirts or obvious heels, etc. Tend to stick with black, grays, blues & browns. Though I have worn white/silver/turquoise/lime-green running shoes -- still working up the courage to wear (in public) some bright pink in those, LOL, as I do have an awesome pair that fits the bill.
Anyway, your call. Good luck with whatever you do!
