Moving is a big step, but the Advocacy process is a giant hassle as well. I think you are on the right track to explore both possibilities. I think talking to the local school is worth a try - particularly because moving may take a year or more.
Also, be sure to visit actual functioning classrooms. You'll learn SO much. Consider renting 'studio space' in the neighbor district if that would qualify you so you can try before you buy.And explore the have your cake and eat it too option by asking the tester to ask if you can attend from out of district. Paying 5K a year may be cheaper than moving.
It may be that the local school would be 'pretty darn good' even with the tiny bit of pull out program. The fact that there even IS a neighboring district with a full time gifted program suggest to me the possibility that you live in an affluent area.
So here's the main question - is the school you now live in, in a so called 'SuperZIP'?
Read this article to see what I'm talking about and ask yourself if this applies to your school -http://online.wsj.com/article/SB10001424052970204301404577170733817181646.html
Bottom Line: the economic class divide in the US has grown so much since the 1960's and I not only dislike this, I fear for the future because of it, BUT being in a SuperZIP community can work really well for kids with IQ 130-140. The local pull out itself might be totally inadequate, but if the school also cluster groups the pull out kids during the rest of the day, and the teacher are used to having several kids with 130-135 IQs in their classrooms, you have a much greater odds to having your child's needs met without having to move. So that is something to consider.
(For the record, it's also true that working class districts can sometime be terrific for VG (very gifted) kids because they are more open to the parent's requests for gradeskips and independent study options because they are keenly aware of their limitations. For kids with over 145 IQ this situation can be even better than SuperZIPs because apparently there will always be kids who are 'too different' some because of twice exceptionalitiy, some from personality, some because of High LOG (Levels of giftedness)
So ask yourself questions like - how many schools does the neighbor district pull from to get their self contained classroom? Is it about the same number of kids that end up at your local High School? For some areas, self contained classrooms are a way to get the kids who are fated to be in those AP classes together for high school together sooner. For other areas the self contained classroom pulls from a population 10 times higher than your local high school and represent an opportunity that is locally rare. Either way a well run self contained classroom with the cut off of 140 is a pretty cool opportunity, but in judging 'how much worse your local option is' the distinction is meaningful.
What are the average SAT or ACT or State exams scores in both districts?
(If they are similar, that means that you have a similar student population to start with, so if the self contained classroom is from a group that is similar in size to your local high school, that your high school will have similar kids.)
I'm summarizing a lot here, years of thinking about this topic, so ask questions if I've lost you, ok?
Love and More Love
Grinity