The 4L60e is probably the most common swap transmission in the off road community. And they put it in eleventy bazillion GM cars. Rebuild parts are plentiful, and even upgrade kits are an easy find. The guts are basically the same no matter the bellhousing.
If it were me, I'd just pull it, take it to the transmission shop, have them do a rebuild with common inexpensive upgrades, and stab it back in.
When you do the TDI swap, that's when swapping a different transmission in would make sense.