So, I pulled it apart, removed the offending socket and bridged the link to make a permanent connection to my internal antenna. I'm back where I started signal-wise, but it's better than not having a phone.
While I was in there I noticed three tiny (0402 size) solder bridge jumpers on the main PCB marked H, E and A. On my device, the A jumper is shorted. Now this is a long shot, but it gives me a theory. My phone was bought in Australia, but last year I moved to Boston. The quad-band frequencies of the Australian GT-N7105T line up just fine with AT&T for phone and 3G/HSPA services, but the LTE service here is on a different band, and I've been living without LTE. Is it possible that the A stands for Australia, E for Europe and H (somehow) for nortH america? Could it be that those jumpers select frequency on the radio, rather than having a different radio module for each region? It makes sense from a design POV - I'm a hardware engineer, and if I could do it that way rather than having totally separate BOMs and SKUs, I would.
TLDR: Can I turn my Australian Note II into an American one by changing a solder bridge?