I don't see the problem. You're moving to a more walkable (bikeable?) neighborhood, and you can still bike to work (either the 18 mi or with transit), so overall you'd probably be able to do less driving around (and since you're a mustachian, you would :) ). Smaller house = smaller bills (heating, cooling, lighting). No yard would be a dealbreaker for some but it's a plus for you... You'd be paying a premium (same price/smaller house), but that's payment for the huge jump in school quality. At least, the measured kind--maybe go visit and check them out, first.
My parents, two younger sisters and I lived in 2BR, 1000 sf or less apartments until I was 15 and were quite happy (then we moved to a 3BR condo that was badly laid out and a pain for 2 years, and they downsized to a 2BR again when I went to college). They could probably have gotten a bigger place on their budget, but the schools were a major factor. 1600sf should be a TON of space for your family of 3-4.