Since you have an L36 Series II 3800, you probably have a 98+ car. Thats going to force most things to be custom.
TH400: Heavy, inefficient, but dang near bulletproof upto about 450-500 from the factory. Utilizes an electric kickdown and vacuum modulator for line pressure and downshifing. Factory equiped with lockup converter on some later models, TH400C.
TH350: Lighter, more efficient, but known to break at relatively low power numbers if not fortified. Utilizes mechanical kickdown and vacuum modulator for line pressure and downshifting. Factory equiped with lockup converter on some later models, TH350C.
TH200-4R: Moderately heavy, moderately efficient, has overdrive, designed to take a moderately modded LC2 from the factory but not known for its unmodified reliablity above 400hp. Utilizes TV cable for line pressure and downshifting. Factory equiped with lockup converter.
I'd stick with the TH200-4R, none of these trannies shares a crossmember with the stock 4th Gen 4L60E though there are crossmembers for 3rd Gen TH200-4R and consiquently TH400. There is also the issue of replacing the vacuum modulator with somethign mechanical though there are some tricks to make the vacuum one functional but not work very well. The other option is full manual valve body so you have to control your own shifts. If you plan on spending any significant time on the street you will want an overdrive as these engines do not like to be spun very fast for very long due to the shortcomings of the stock oiling system. Also you will likely need a better rearend as the stock one might very well give up the ghost with the LC2 torque curve and any sort of traction.
Just some info,