Why yes! you can get a few types of Limited Slip for a few different transmissions that bolt to the A series engines. Most any trans that bolts to any A engine will swap between them, so the 5spd from an MR-2 (4agze) will mate up as will some other japanese market options from the corolla levin (4age engines) which either come with or have available aftermarket a LSD. Mechanical, clutch based, etc. You might be able to get more information on that by hitting up 6gc.net since the 7afe came in the 6th gen celica as well.
110lb/ft at 2500rpm on a 500-700lb buggy is just going to sit and spin even with a limited slip if you dump it from a stand still, I can't imagine that thing on street tires on the 1/8th mile...oh wait...yea I can. FUCKING FAST.
Another option would be a later model Toyota engine...the 1ZZ-FE. its even lighter than the A series (1zz is all aluminum) and makes more peak hp and peak torque over a wider rpm range.
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Output is between 120 hp (89 kW) at 5600 rpm with 122 ft·lb (165 N·m) of torque at 4400 rpm, and 140 hp (104 kW) at 6400 rpm with 125.8 ft·lb (170.6 N·m) of torque at 4200 rpm
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the later models of the engine have VVTI which boosts not only power output and smooths torque but enhanced fuel economy. the 1zz I think also mates to many transmissions the A series mates to as well and would also be an excellent choice. peak torque comes in a little later but over the whole RPM range its got more power, so at 2800 RPM its got only a smidgin less than the 7A-FE or about the same, but the power keeps coming as it winds out.
I don't know how much the 7A-FE weighs, but the 1ZZ-FE weighs about 225lbs. I suspect the 7A is closer to 350.