Sure. I have a Quaife LSD in all my cars now as I just put one in my 100HP. They are good, but few people will realistically do this for a bit of fun on the track. People are happy to spend a fair bit on wheels and coilovers because it achieves a look they want with a by-product of performance in some cases.
I said gripper tyres will help, but not a lot. My classic Panda will still not put full or near full power down out of tight corners, even with the LSD and semi-slick tyres. It is several times better than without the LSD, but still must obey the physics when on the ragged edge.
I think it is easy to come up with the best solution, but an easy or "right" solution is sometimes a difficult judgement. Sometimes that point is missed quite a bit in my opinion.
As suggested, a cheap softish tyre like the T1Rs is a pretty good way to go for this guy. The R888s in most places are a bit rich for the gains perhaps. Second hand racing slicks are good track option in the dry at around £10-£15 a ago.