I think the stock auto is pretty tough. I wonder if it's the same as the SR20DET transmission, which is obviously boosted from the factory. For the manuals, the main difference is the bell-housing and the internals are the same. So, I wonder/ hope that the same could be said for the automatics. Anyway, we have a sometimes overlooked resource here called Articles. Here's some links to a couple and you'll notice on the bottom of each article are more links (in red) that lead to similar topics!
http://www.nicoclub.com/archives/turboc ... 240sx.html
http://www.nicoclub.com/archives/nissan ... sions.html
Maybe you just need a torque converter, maybe you'll want a valve job and a rebuild, maybe add a tranny cooler. Sounds like you're simply running a KA24E (or is it DE?). Power would be maybe ballpark 200HP? I think you'd be fine with the tranny you have.
Ooh, I always thought this article was interesting, maybe you will also.
http://www.nicoclub.com/archives/240sxp ... rsion.html