Not sure what the thing in the middle, just to the left of the oil filter is, but it leaks oil when the car runs. Looks like something related to the oil pressure relief valve and/or cooler supply, but nothing directly called out in the ROM that I could see.

Looks like it's gonna be a beast to fix, so any help is greatly appreciated.

 

Which bit precisely do you mean? If you mean the big casting, there is an O ring seal in there which fails, so if you unbolt the casting you will find it and easily renew it.
See parts 18 amd 19 here:
https://parts.jaguarlandroverclassic.../brand/jaguar/

 

The 'O'rings in that unit leak. I replaced many for oil leaks in the oil cooler system.
Unbolt the pipe and the top and bottom fasteners, collapse the fittings, slide them out and replace the 'O'ring seal.

 

Cool, thank you. How much oil should expect to come out when I take that apart?

 

I think the oil will 'bubble-up' from the oil pan as the level is a little higher than the bottom fitting.
It's been about 30 years since the last one I did.

 

very little

 

First pic is what a "very little" looks like 😁. Looks like a quart or two. Last pic shows the nut you need to get first.