Cricket, does she know that she is working harder than others do at things like reading, and if so, what explanation does she make for herself? Does she see that she's struggling, or not?
If she doesn't see it, then it makes sense she wouldn't want a dx; it's an explanation where no question was asked. She's also at a tough age-- nobody wants to be different at 12 or 13.
When she gets a so-so grade in the GT class, does she chalk it up to not being GT? I can certainly understand why this is a concern.
Having your DH do the talking sounds like a good idea. You could also level with her about what her test scores are and what they mean; if you think she's prepared for that kind of reasoning.
For my DS (younger than your DD) finding out his dx was important for his self-esteem and self-awareness-- it gave him an explanation other than "I'm not good enough to succeed." I hope your DD ultimately gets to that place too.
DeeDee
