I put a remanufactured tranny and torque converter in my E53 4.4i a few years ago from Euro Transmission Specialists. I had it shipped to Louisiana, but I seem to recall they are in LA (310 area code at any rate). I got an off the shelf one from them and sent them my core.
My transmission had been slipping for several years, starting at around 110k miles or so. Sport mode helped quite a bit for a while so we rode it until failure (it was my kids car at that point), which it did on my daughter one afternoon, conveniently a few blocks from my house.
I did some research and settled on Euro Transmission and have been pleased so far. No drama so far, going on three + years. The 4.6 gearbox is the same basic ZF model, but a bit different as it has an extra clutch and a different stall converter so I'd make sure they can get parts and rebuild yours. I spent some time looking for a 4.6 TQ for the engine swap I'm doing and they seem to be very hard to come by. That said, guys who have done the 4.6 swap into a 4.4 car (5 series or e53) will tell you that the car will run just fine with the 4.6 and the 4.4 ZF, in case you have to go with a 4.4 TQ for some reason.
https://eurotransspecialists.com/contact/