Update: In the field today (fortunately, with no client watching) I went through a similar hair-pulling exercise trying to get everything working. The T2 and the radio appeared to be talking, but To Base wasn't working, as evidenced by the radio call sign change not getting to the radio (i.e. make changes, press To Base, wait for completion, press From Base and see that the changes didn't stick). After an hour or so of getting nowhere I finally called Shawn, and a combination of his Texas drawl and a hard reset on the T-LS made everything sing. Soon I was reaching out 4 miles and hitting points in my network within a few hundredths H and a tenth or less V.
Well, almost. When I staked out the network points I was actually missing them by over 4 feet horizontally. I figured it was a datum issue, but I was tired of not getting anything done and I didn't have to do any stakeout, so I just kept going. When I got back to the office, I saw that the base point broadcasting reference frame was set to WGS84, while my network is NAD83(2011). I know I had set the base point frame to NAD83 at some point, but I guess in all the setup iterations it had reverted somehow. (I'll have to keep an eye on that in the future.) The gfile export allowed me to fix it all in Star*Net, so no loss.
P.S. It looks like the T2 or J-Field isn't able to read the power output setting of the HPT435BT, because the T-LS never did show anything more than "---mW" in the display, even when I was getting corrections.