Should do - I fitted a 4EFTE gearbox to a E11 Corolla 4EFE with no real hassle.
I had a look at an E11 Corolla box I have spare (forgive the state of it):
View attachment 5428
The bell housings (the large bell shaped part that bolts to the block) look to be the same.
View attachment 5432
Two things of note..
The speedo gets it's reading via a manual cable on the 4EFTE gearbox which screws in here:
View attachment 5431
Your car uses an electronic version so that would need swapped:
View attachment 5430
I can give this a whirl to see how easy it is, never done it before so worth exploring.
Also challenge number two is that your engine has a fourth mount on the front of the gearbox that the 4EFTE doesn't have:
View attachment 5429
This will probably get in the way of whatever turbo kit/downpipe you are going to run. There might be some custom fabrication involved which no-one except the guy doing the swap will be able to sort for you. If it were me I'd try running without that mount and see how much it affects things. All our Starlets are running bigger power than the 4EFE and we all live with three engine/gearbox mounts. Can't see it being too big a factor.
The rear and side mounts could be swapped from your current box to ensure fitment to your car. The bolt holes seem to be in the same place as the 4EFTE housing.