OP, by all means, rent a scooter to meet your needs.
I should not post here...
But we've been in your shoes. It's sort a easy to say you should shouldn't care about rude comments, but another thing entirely to be in the middle of them.
When a very healthy family member (also very slender) suffered an injury a few years back, mostly using the scooter was great...During some moments it was a trial.
It wasn't just other guests that made comments The shocking thing was that some CM's were unkind. More than once, CM's refused to permit her to enter the handicapped queue. They told her she HAD to park it outside the ride. I mean, more than once we entered a ride right behind another patron in a scooter, and they were not asked anything. We weren't in a mood to fight, so we just did our best. It was pretty clear the CM's she was faking. It was a bit un-magical.
On subsequent trips, we've seen folks struggling to maneuver them. One tricky spot is the monorail. The spacing can be tight, especially if they have to load two scooters in one car. I recall one woman nearly in tears because she just couldn't figure out how to make hers go the way she wanted. Knowing the monorail was loaded, waiting to go....We assured her she was fine, and tried to help, but it was quite clear, she was close to tears. Nobody was rude, it was just a tough situation.
My family member too, also found that her injury made her more tired than usual.
Oh, and one more thing about the monorail- Make sure the monorail operators know where you want to exit- before you load. If at all possible, when you reach your stop- have your daughter play advocate. A CM has to carry over a metal ramp before scooters can unload. CM's are supposed to call ahead, but the system is not 100%. A few times we ran into issues. The best bet is to have your daughter kind of stand in the doorway kind of waving her arms until a CM arrives with the ramp. Otherwise, if you miss your stop, you might have to go around the whole monorail loop, which takes quite a few minutes. And then you'll still need someone to flag down a CM! I'm sure plenty of folks scooter on the monorail solo, but we found it useful to have someone with the scooter.
Still, we found the monorail was easier than the buses.
Our crazy solution we stumbled upon, because my injured family member could stand for very short periods, was to sometimes discreetly swap. When an older family member got in the scooter- nobody gave us flack. Truly. Though the elder member in our family was perfectly healthy. It was only when our injured family member was using it that we got some looks. So sometimes the best solution was to let the older family member retrieve it.
This might not be a solution that works for this op, and others may hate it, but it helped us avoid the emotional drain of dirty looks. Which made our visit better.
I think adults, like children- can sometimes get over tired, or over stimulated at WDW. If anyone in our family has an injury, it helps if we don't try to do too much. We take more breaks than we normally take, we took a slower pace than we normally take, and we see more shows than we normally see. The lady I described on the monorail confessed that she was tired. A family ember explained it was arrival day, and they'd traveled a long distance. They were returning from dinner at GF.
So maybe make a pack with your daughter. You'll do what you can do, but when you say you've had enough, it's her job to respect your wishes. It might be a good way to preempt disappointment.