MT is less forgiving of an unresponsive engine, as the engine is directly coupled to the rear wheels. AT with torque converter will allow the engine to spin up a little and feels less responsive to begin with but can mitigate that effect by allowing the engine and turbos to spin up when you nail it. I remember back in the early days of turbos in road cars some writers would actually say that they preferred ATs with turbos because it could help mitigate lag (remember massive turbo lag?) I never drove one because back then I couldn't afford anything beyond "runs."
As to the LSD, everyone I've talked to who has one says it makes a world of difference to the car, but you're looking at $2K or thereabouts for a good one, and they're not easy to install (apparently the ring gear is welded to the carrier on most E9x, so this is not a DIY process other than just swapping the whole unit.)