The 5HP24 transmission is used in both the 4.4 and the 4.6. There is no part number for the bell housing portion for the transmission. What info do you have that indicates the torque convertor won't fit?
Assuming the 4.6 torque convertor fits there are lots of other specs particular to the application that enter into choosing the appropriate convertor for your needs. The following explains the complexity of that chore:
https://www.youtube.com/watch?v=j5JFMs8gdbs
If it doesn't fit the option is out the window. If it does, you may or may not like how it changes the driving experience, assuming it is enough to notice. We would have to know the rpms that the 4.4 and 4.6 cams 'come on' to know if that matters but diff ratios are different, tire size is different etc.
IMO, based on the low frequency of 4.4 torque convertors failures, the stock convertor is all that you need in a stock 4.4 even if you are towing.
It is important to reiterate that at this point none of us know the cause of the problem you are having and IMO taking it to a 'certified' well recommended transmission shop familiar with BMW transmissions before spending a dime on new parts or tearing it down is the best course of action. If I were to spend any $$$ it would be to drain the fluid and remove the pan so you can see if there is metal in the pan, examine the fluid, replace the filter and fluid to see if there is improvement. Quite often, once there is a transmission problem changing fluid will at best be a short term benefit but if is worth a try.
Be sure to make an informed decision before you decide to pull and repair an automatic transmission, it's not for the faint hearted.