Well, you've already tried the front-door method of simply asking after doing your homework and looking at the obvious means.
In light of that, I'd say-- do more focused research into WHAT your options are locally, now that you know that you're not likely to get much help from the power-that-be in doing that investigation.
As a first step, find out if individual schools have GT titles anywhere in their info-- and make phone calls more directly to people in administration at individual schools.
It might well be that neighboring districts don't want students who are the reason why your district isn't making AYP, if you see my point. A GT student who is VERY likely to boost pass rates might be another story entirely and be worthy of an 'exception' to such a policy.
You won't be told that by district officials, no doubt, and you won't be told it at all officially, that is my guess... so ask about "exceptions" for extraordinary circumstances, etc. You may have to give people some space to go outside the lines a little. See what happens.
Mostly, look at the individual school level-- find out what you can and then make phone calls to get additional information about programs, etc. at individual schools. Have a short list of questions that matter most to you.
Then you'll know WHAT, and at that point, you can begin finding out what it will take to access any one of those options.
