I do not think the airline matters. Money is money however you are dressed.
Toilets on planes are unisex. One you are on the plane, I do not think any issues are likely to arise.

The issue is with security checks. What form of picture id are you showing at security for travel. If you have a valid form of acceptable ID in your alternate persona. If you do, then security checks are not an issue. If you do not, then you are likely to have issues. Not sure how you would get around those.