I guessed the polarity wrong at first (the relay datasheet didn't give any super obvious indication as to which of the two directions of the twin-coil relay was going to one port or the other).
But on the second FPGA bitstream I got all of the relays switched to output mode and I have nice happy pulses coming out of them.
So now that's all 12 output channels (4x unbuffered 1.8V output, 4x buffered output, 4x buffered bidir) confirmed working.
That's probably it for tonight since I have some other stuff I wanted to get done outside the lab, but good progress.
Tomorrow evening my focus will be on verifying the input ports. At that point I'll have all of the building blocks I need for minimal functionality of the core trigger-crossbar functionality controlled over serial console.
Then I can start thinking about debugging the power issues more.