Well, I'm just going to pipe up as a proud citizen of Birmingham and encourage you to consider living in the city itself. It's true that Birmingham City School system struggles, in part because most students fall into categories that traditionally struggle, but if your children are smart and academically oriented, they could probably get into some of Birmingham's magnet schools (WJ Christian, Phillips Academy, Princeton, EPIC, Ramsay, Alabama School of Fine Arts, JCIB), and Birmingham's magnet schools are on par with the best suburban schools. There are also some great private schools -- Alabama Waldorf School, Red Mountain Community School, Advent, Altamont--in the city of Birmingham.
Where I live in Crestwood North, I can drive my daughter to her private school (Red Mountain Community) in less than 5 minutes, we can bike or walk or take public transit if we're up to it, I'm less than 10 minutes from the airport without having to deal with the interstate at all, but most of the houses are a bit smaller than what you're looking for. (There are a few lots available, though, if you want to go the build-what-I-want route.) That said, you could easily find a house that size in Avondale or Forest Park or Highland Park or even Norwood--all in the city. You can check out some of these Birmingham neighborhoods, and others, at http://livbirmingham.org/index.php . Good luck!