I wouldn't worry too much about, take her to whatever is the cleanest toilet that is free when she needs to go.
You are lucky in that as long as you have a quick look in the door of the ladies and it is either occupied by ladies or completely empty, you could at age 3 quite happily let her go in there herself. If she gets in any difficulty anybody would help her. Women using the ladies will be in cubicles and in most cases would not mind you popping your head round the door to check on her.
As the mother of 2 sons who are now getting older, I find it more difficult to take them into the ladies, yet unless I can send the two of them together, I am not comfortable with sending my son into the gents on his own. My boys are 7 and 6.
I had one nasty woman complaining that I had my 7 year old in the ladies with me - granted, he does look older, but I did confront her about whether she would be happy to send a 7 year old girl into the gents on her own, and in my book, he is as likely to be abused in a gents toilet as a girl would be. She did shut up, but as everyone was in a cubicle, I dont really see what the problem was.