Now, when you say you can turbo charge the B16sirII does that include pistons/rings, rods, valvetrain, and other related machine work? If not, get the b18b, much better for boost out of the box. You could later do a LsVtec change later, but you'll be better off with this motor. Now, you could buy the already vtec b18c, you get .2 more liters of displacement, but again, you really should build it and it costs somewhat more do build this engine than a b16. As for the H22. (handling) Sure, it will handle well, but at a large suspension investment, that includes having your coilover set up properly tuned by an experienced technician, not yourself. But, put that money in a car with a B series engine and the H22 will loose ground every turn. Still a very good swap though. (Engine power) An H22 can run 13's all motor with bolt ons and good driving. If you want to go turbo, its pricy, you HAVE to build it. the piston design puts the rings high up on the piston and they fry. sucks. But if you could gather the money, it would be very very powerful with good tuning.
Without knowing what kind of compromises your willing to make in cash and performance, its hard to really answer your question. From what i know now, I would go with a B18b(later vtec hybrid if you want vtec). You could build a B20 turbo which is a more difficult option but a good one.