I don't think that treating people poorly because you're worried that not doing so could hurt your career is the right thing to do. It might be pragmatic in your case, but so is taking money out of a wallet you find on the ground.
> I lost my best friend due to a lie I couldn't disprove.
You lost your best friend because your best friend didn't trust you and because someone lied. None of that is your fault. Sometimes the world is a shitty place with shitty people. That isn't a reason to add to that shittiness by refusing to treat women in the workplace as you would men.
I've returned every wallet I've ever found and I've never taken any money from them. Doing so poses no risk to myself. It's very easy to stand on your high horse and criticise others but after you've been in the situation of being accused of something it changes your opinion on certain things. I hope that you never have to deal with it.