That’s a transmission crossmember but I don’t know the exact nomenclature used in the Mopar parts system.
Regarding the transmission pan, if you use a jack you can unbolt it and lower it carefully, taking care not to spill any fluid. That will allow you to inspect everything inside on the bottom that you can see and check the fluid to see if there’s any small broken parts (I doubt it). Then you could reuse the same fluid and just bolt it up, or replace it with the exact amount of new fluid in the pan since there isn’t a filler neck up top.
Here is a Poison Spyder one for the JL. They have a video of installing on a JK but I imagine it will be similar for the JL. Not cheap but not super expensive, either. Looks like a worthy upgrade to stock.