D.P.
I had a few thoughts when reading your post.
First, could you try a sensory training-type thing? Just going INTO the public restroom. Next time going into a stall. Next time try to pee on the toilet?
Secondly, all of us moms know that we can NEVER get our kids to do something that a complete stranger easily could. Not that I'm saying to get a stranger to take her to the bathroom, but will she go at a grandparent's house, an aunt's, etc? What if her dad took her into a family restroom? Do you think that might work?
I know how you feel...my son just hates to take the time to pee. He'll get up in the morning after holding it all night and would wait hours to pee if I didn't tell him/remind him to go. He seems to be able to hold it forever, but I'm sure that can't be a good thing physically. Good luck to you!