Well, see, that's the thing. If most people can't get licenses anyway, it's not as big of a deal, y'know? Where I'm from, you get a permit at sixteen and then a full license a few months after that. Additionally, because my area is very reliant on cars (so there's not a great bus system and so not a lot is within walking distance), it's very unusual to have someone who doesn't have their license or doesn't drive that often. Granted, you have a point in that society differs from area to area, but the main point I was making with the analogy is that getting a Pokémon trainer's license is a lot like a situation where it's unusual for a person not to drive. As far as we've seen in canon, the society pushes a person to get a license at ten because it's just strange for someone not to -- as in, most other people who intend on being trainers get their license at ten, and it's not exactly a frequent occurrence to see someone who's older go out for one.
With that being said, it depends on how you handle it. As I've said before, there's really nothing wrong with the age itself. It's all about how it's treated and why her parents decided to hold her back. I haven't actually read your story yet, and I can't quite grasp the full details of her situation just from what you've told me. For that reason, I don't know if you've actually explained it well. All I can say is that it could go either way. It's possible that if you took care of the more pressing issues, that one will stand out more (if, of course, it's a shaky explanation), or readers still won't mind (because it's at least not as drastic as aging up a trainer to fourteen or sixteen for no apparent reason). For the sake of avoiding shaky narration, if you're going to bring it up at all, it's best to err on the side of caution and make sure that part's strong (read: well-explained) as well. After all, you don't want to bring any fact into a story without having a reason for presenting it to an audience, which is my main point, really. (As in, if it's important for her to be eleven, that's fine, but you don't want to mention this if the reader doesn't absolutely have to know how old she is. Otherwise, you'll be implying that there's some important reason why she's eleven, hence the "explain it in detail" advice -- because then you'd be obligated to make sure it makes absolute sense to a reader.)
On that note (although I might be repeating myself by now), it's actually perfectly possible to go an entire story without mentioning how old a character is at all, especially if it's not particularly important. Have you thought about that? I mean, if you're really concerned about whether or not your reader would be put off by which parts of canon you're bending.