the lsd connects to the shaft coming off of the transmission.. it has nothing to do with the transmission itself.
the type r tranny will make your car slightly quicker, but it will also have you cruising 60mph at around 3.5k vs 3k... and it will lower your theoretical top speed.
the 6 speed type r transmissions are from the dc5 type r most likely, otherwise they are aftermarket transmisssions or something

if you want just the lsd and keep your gsr tranny you can.