I applaud you (loudly!) for the many ways you're working to ensure that girls all get to know each other and don't form cliques within the larger group. The idea of mixing girls up as competition buddies is a good one, as is mixing girls up for exercises. You are doing well. I agree with you that this sounds like an issue of a very shy girl, not like an issue of other girls targeting her for intentional exclusion. I would agree with your definition of what "bullying by exclusion" would mean, and this isn't it.
Did you tell the mom what you told us in your post here - the part about how she always has a partner in stunts, but she hangs back on her own when they have breaks, etc.? See if you can convey that to the mom (and I hope she won't blow up -- but she's already throwing around the term "bullying" which indicates to me she may be readier to blame other kids, or you, than to see that her own child needs to be doing some work here).
Your suggestion about more play dates outside cheer between this girl and other girls is a good one. I would reiterate that with the mom. Does the mom know that you intentionally mix the kids up in so many ways? I would ensure she does know it.
The only other thing I can think of would be to take aside a girl or girls whom you know well and who are particularly mature (and able to avoid telling all the other girls what's going on) and asking that girl or those girls to make an effort to reach out to this shy girl more during breaks at the studio and down times during competitions. If you have a smart, more mature girl who can handle this with sincerity and kindness, it could be a big help.
You sound like you pay careful attention to the dynamics in your group. Good for you!