I was a competitive cheerleader for 5 years. Being 4'11 and 95 lbs, I was the flyer. It was no question that some were more suited for fliers and others as bases. I was awful at tumbling... I left that to the girls who had it mastered, and the team worked around each girls best traits... though we did after competitions and down times spend time experimenting with other spots for fun.
I think reminding her that you are focusing on her best traits for the team, such as her strong base and dancing skills is really where you need her. For instance , in football a linebacker and a quarterback have totally different purposes (and usually bodytypes) and the team would not work as well if they switched places.
It's sad that it has come to this though, that she isn't getting it, even though her bases fall every time they try and get her up... I'm guessing it could be pressure from her mom. Just tell her that the smaller girls are easier to lift because the bases are not that strong yet, and that for safety's sake, she isn't ready yet to be a flier b/c she hasn't mastered being able to come down without flailing her arms and legs, bearing her weight during the take off and maintaining her balance. I use to practice these things and adding in heelstrecthes and scorpions at home on a barstool.
When we first started cheering, we all had to do basic stunts, like the L stands on the thighs and such for safety. We had to do the most basic things and master them and sign off on them on a sheet before we were allowed to advance to the next stunt with a bit more difficulty. Perhaps implementing something like that would be helpful.
It's also about technique, not just weight. We had one heavy flier, but she was solid, held her weight really well in her upper arms and actually flew up fairly easily because she had great technique, pinched her butt and had great balance, though her landings were pretty rough and she had a hard time bouncing out of the holds into new stunts or jumps because of that.
Here is a pretty good response to that to help her with the weight bearing as a flier (scroll down for the answer):
http://en.allexperts.com/q/Cheerleading-2239/Helping-Flye...
If you need to get her up, to say a half extension (no way to try a full with her), you can always throw in 4 bases, with a 5th there holding up one of the main sides. The bases really need to have their bearings and keep their arms close to their chests.
If you do a cheer for a game, let her be in a front and center stunt, even if it is just a little thigh stand like this, using a back base as well, and then you can have more complicated stunts forming a pyramid around her as the fliers connect arms:
http://www.ksd.k12.ky.us/Athletics/images/Back%20to%20Bas...
Or, you can let her do a little dance during the stunt, part of the cheer, maybe with 3 dancers and her in the middle front so she still feels like center stage.
In the long run, if you try all this, she still persists and she's not progressing on the practices, simply tell her you are going to be working on the safer stunts that the bases can handle for the time being. Ultimately safety is more important than hurting someone's feelings. Tough, but that's the way it is.
I fell flat on my back in the middle, halfway straddling my back base who hit the floor of the Mavericks stadium coming out of a basket toss... because the stunt was thrown backwards instead of straight up and when I got out of that stunt, I went straight into my next stunt and finished the cheer and straight into a dance, but I had tunnel vision, pain and couldn't remember any words to the cheer so I just smiled. I can only imagine, an untrained heavier flier falling on a smaller base and how much more dangerous that could be for both of them. I've had back problems ever since.
As for the mom, don't include her as much. She may be nosy and bossy and mean, but try not to let her run the court. Maybe have a parent pow-wow about being supportive and team players, and supporting the girls and that would let her know that you are on to her.