1) Are the transmissions in a 2000 GSR, Type R and a Prelude the same?

2) If so did the R come with an LSD?

I have a problem with my 2000 Prelude 5sp trans not going into reverse after a bad shift from 2nd to 3rd at 7k+.
I have a M2Y4 model trans with an H22A4 motor.

Since I have to pull the trans I want to upgrade to an LSD, but Quaife ATB LSD's are $946 + at Inlinefour.
I'm wondering if there is a cheaper way to go? Any advice?

Thanks,
Doug