"SW20" is the body code for the second gen MR2. Those run 3SGE (And 3SGTE) motors.
The AW11 is the first gen MR2, and it has a 4AGE (4AGZE in the supercharged config).
You should know from being just here in Hondaswap that when you do a motor swap, the trans doesn't matter. You'll be moving the 4AGE over, to hell with the trans.
On the darker side, it won't bolt to the Corolla's transmission. You'll need a T5 trans, and the 6.5 inch rear, with a custom axle.
If you instead consider some already RWD cars from Toyota, you'll have the 5MGE / 6MGE / 7MG(T)E engines, with the W58 trans and rears. Then a custom shaft, and you're good to go. The rear is a longer axle (Read: Wider) so you'll need to play with custom offsets to make it look right.
Put a V8, (Preferably a Ford) W50 Celica trans, and then custom shaft for the stock rear. It can't handle more than 250 lbs/ft of torque, but that shouldn't be a problem.
-> Steve