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That's a pretty serious project- I think someone who had a hand in building Lloyd's car was on here, you might find the odd topic and be able to contact them to find out what they used. The front bit was a Lexus 1UZ, but the rest might be interesting for you. I'm with Littleredspirit in that you want the lightest gear that will just handle the power. Even replacing the 4K with a 4AGE upsets the light steering and handling a stock KE70 has. Ah- here's some info for you.

If your goals are 4age, then you wont need a more hd diff than a T unless you turbo it, and it would present a performance disadvantage to have a big heavy diff for the small low torque high revving 4age to drag around. Id lean towards having a T diff gear set cryoed for strength as opposed to going bigger. AE86 housing will fit and be an appropriate width. They do have terrible disc brakes, if you want a low fuss handbrake that locks up use a t18 or early ae86 housing that had them. Or you can start with a disc housing, upgrade to s13 or U13 rotors and go to double caliper brackets.

You can use a housing from a t18, or ae86, or early te71 which you might have access to in NZ. Buy gears from wherever if needed, Techno toy tuning, weir, toyota etc, LSD centres are available, I can highly recommend the trd 2 way, but be ready to compression lock it if you cant heel toe at all, or be ready to learn to heel and toe as its pretty easy to get the feel of with corolla pedals. If thats a big ask, look for a 1.5 way, so it wont savagely lock in both accel and decel. Most t series diff are small axle zenki type, with approx 24mm axles. At some point around mid 85 Toyota swapped to a larger axle size and double row axle bearing in the ae86. This is called late or kouki. its faintly stronger and more reliable, and you need to make sure your lsd is compatible with the series of diff you use. Some axles will need around 5mm lathed off them to fit into the lsds, some wont, it depends on the brand of lsd. You can source lsds for t series toyota from cusco, trd, tomei, or maybe OS Giken. As I say, the trd ones are excellent, I havent owned the others, but the two different trd 2 ways I owned were tops.

Yep- Write out a plan for what you eventually want to do to the whole car, then work out what you need.. T-series diff, LSD, T50 gearbox, driveshaft to suit, 4AGE motor, and stuff, and start looking for parts. It will take a while to assemble everything you need.

K40 is the smallest gearbox. You could probably fit a t50 from an ae71 or ae86 and upgrade to hydraulic clutch.