I have RA and have used an ECV at WDW in the past- and I must be honest- it was really wonderful- but I was in worse shape then. And it did have its drawbacks, primarily the getting it to and from the parks. Also, it was pretty inconvenient when it came to rides (you probably know this, but having an ECV or wheelchair DOES NOT make it easier/faster overall to get on rides or see shows).
So, this trip, I'm not going to rent one in advance. I know all the arguments pro & con, but this is my choice.
I do think that you have to honor your daughter's feeling about the ECV (as misplaced as it might be). You've been to WDW before, so you know what you're up against in terms of walking. But it's really hard having a chronic illness and it must be even worse being young and having this problem. Give her the facts and options, and let her decide. She'll be having to make these choices her whole life long, might as well start now. I know it's really hard as a mother to even take a chance on seeing your daughter in pain- my family was very happy that I had rented an ECV. None of us who have a mobility issue want to disrupt anyone else's trip if we can help it. These are really hard choices so there's a lot to consider, I'm not sure there is a right answer except to give it a try one way or the other and be ready to change course if it doesn't work out.
If she decides against renting an ECV for entire trip, you should have a back-up plan (as do I). If they run out of
ECVs at the parks, are there plenty of healthy folks in your party who can push her if only wheelchairs are available? Will they mind doing so? Have you checked with some local companies to see if they can deliver an ECV on short notice if you decide to rent one part-way through your trip? Does she have access to extra pain medicine if she needs it? Do you have a handicapped placard to use if you have your own car? She may not have needed any of this before, but little things can make a big difference at WDW. One thing I've learned to do is to sit down at every opportunity- even if I feel fine. Because it all adds up over the course of the day, and by the time she feels bad it may be too late.
So if she doesn't want an ECV, there are lots of little things that can help, but be ready to change plans if it doesn't work out the way she hopes. Good luck!