Trimming a power supply to 0 V output in current limit
Right now, when the device changes to a current regime and turns off the output, I'm seeing an output of a little more than negative 0.5 V.

That seems very unlikely.

My reason for saying that is that the output transistor can only connect to the rectifier positive supply.
The only thing I can think of is that that Q4 is turning off completely, U2 is going negative and drawing urrent through R7, D11, D10 and R15. The output voltage would then be clamped at -VF of D11 (as R7 is only a half ohm shunt and won't drop much voltage).
Try monitoring the output of U2 under current limit conditions and see what's going on.