Today is a good day!
While I did manage to get a mouthful of coolant, I also managed to fix the oil pressure problem!
Yep, it was the oil pressure relief valve. Had a tiny bit of metal in it, a bit bigger than a grain of sand, holding it open.
Here's what I had to do to get the pressure relief valve out:
Tools needed: 10mm wrench, 10mm socket, 13mm wrench, 13mm socket, 25mm socket, wobble socket extensions.
Loosen up the upper hydraulic line first
Loosen up the lower hydraulic line next, starting with the upper part. See all the green fluid? Coolant line behind the hydraulic line is in the way. Unless you want to change the coolant, don't remove this line!

A very long 1/4" socket extension is required for the back bolt. I used a 9" one.
To remove the oil pressure valve, you need a 25mm wrench. I don't have one, so I used a 1" (25.4mm) wrench instead. Note the wobble extension is absolutely required here. I also removed the oil dipstick tube just to get it out of the way (10mm bolt)
Valve loosened
The offending part, after removing the piece of metal