Depending on what your dd wants to do, I would go ahead have her eat lunch with the older kids and then revisit the grade skip issue with the school. If she's doing well at lunch, then they've totally lost any credibility for the "social difficulties" argument against it. And by their own doing.

Something like, "Well, I know your concern with the grade skip was the challenges it might create for her socially, so this lunch switch has been a great way to test that out. Now we know she will do fine, so let's get that grade skip going!" wink

Or maybe she does have a hard time, and then you will know that too and can respond accordingly. I would suspect, however, that if she is already doing GT with the older kids, she wouldn't have too much trouble.