What transistor or combination of transistors should I use to make a fully controllable AC switch?
Yes, hefty back-to-back MOSFETs (maybe) or IGBTs should do what you want. Driving them would require isolated gate drive circuits. The complexity will vary a bit depending on how quickly you want them to switch. IGBTs tend to switch a lot more lethargically, but are more suited for the ~1200V PIV type operation required in this application. 

SCRs (and triacs) as, you point out, have to wait for a (current) zero crossing, and the high frequency operation you speak of (20kHz) means that they would tend to stay on due to the high dv/dt at the zero crossing.