Hmmm...
If being in that teacher's class would be bad for your DD, then I would say let GATE go for this year and take the CogAT next year. FWIW, I think the teacher matters more than anything else. A good one can make a lousy situation work beautifully; a bad one can tank even the most perfect arrangements.
I think I'd either ask the principal why she's negative about GATE or disregard her negativity. Without more info, you are REALLY flying blind on that one, and as you say, there could be so many reasons why she's negative. Could you call her to ask, or is that out of the question?
Also, have you talked to any parents whose kids are in the GATE program? If you can, I would. And talk to the kids themselves, too. This can tell you a lot about the value of the program for your DD. Just try to get more than one opinion if you can so that you're not just getting one person's bias. But one opinion is better than none, I think.
For your son...not to sound like the home schooling pusher or something, but is home schooling a possibility? I ask because it's pretty much the only major educational option you didn't list as a possibility for him.
I won't go on and on about it, but if you are even remotely interested, please let me know and I'll be happy to talk pros and cons with you so you're not flying blind there at least. Many people reject home schooling because they think they'd have to replicate public school in their homes. Nope! It's much easier than that! Especially with a gifted kid, it takes a lot less time and effort than you might think. (Frankly, it's probably more or less what you're doing already when you talk about "providing enrichment." That's called "afterschooling" in home schooling circles, and it's one way for working parents to homeschool kids--use school as free child care and do the real teaching at home.) Anyway, a couple of hours a day of official schoolwork is plenty for us most days, and my DS still covered a semester of math in less than 5 weeks. (BTW, I slowed him down to that speed!) We do science experiments, we read about whatever interests him, we "play" with mazes and tangrams, and he has a lot more time to spend with friends. Heck, when he was in public school, it used to take us over an hour each night to get him to do his (way too easy) homework! Mostly that was negative nagging time, too. Now the work he's doing is virtually all pleasant and developmentally appropriate, so it's fun for all of us!
'Nuff said. I just wanted to throw that out there because I do think you're probably going to have to do something for him. He sounds like a fun kid!
