My youngest ds had toileting issues for a long time. For us there were different reasons, and it was bowel not bladder. It was a long time before he was completely toilet trained and off of medication. So, I sympathize with your concerns. That said, he was also very sensitive to accidents, despite being constantly reassured that it was just his body and beyond his control. For him, this would often lead to him withholding, which caused all kinds of other problems. Since your dd has kidney issues, you REALLY don't want that to happen. And from what you have said she find accidents to be very upsetting.
I would first do my homework, then have a talk (or series of talks) with her. By do your homework I mean that there are many different types pf pullups and pads. If you were to go to a few cloth diapering sites, there are many cloth options that do not seem at all like diapers. There are cloth diaper inserts or doublers, again they don't seem so much like diapers. There are also cloth menstrual pads, which could work if it's not a significant quantity of liquid. Any of the cloth products would be breathable, and much less likely to lead to an infection. Then there are cloth training pants, like a pullup, but cloth. Any of these might not seem so much like a diaper to her.
Once you get some ideas, then you talk to her about them. I suspect if you phrase things the right way, you may be able to get buy in from her. If you can get buy in from her, and she understands that this is just insurance in special situations, it could work out well for her.
All of that aside, I think you will be okay. It is worth talking to guest relations to see if there is anything they can do for you. You are going at a slower time, which will be a huge help. I would look at ride info online so you can get an idea of what the longer rides are. For example, Ellen's Energy Adventure is quite long, and once in, you are stuck. Like everyone else has said, hydrate and lots of potty breaks. You could even make it a game, how many bathrooms can you visit per day

I haven't been since the new Fantasyland Expansion opened, but the Rapunzel bathroom is supposed to be beautiful.