The board plugs into the instrument with a ~ 60 pin connector. As you say, I could just wire out the 15 V regulator node, but it's not that which is bad - it's shorted to GND somewhere.
Edit -- I fixed it. There was a short on the PCB somewhere which shorted the 15 V regulator output to GND. It has a thermal shutdown, but must have eventually given up. Removing it still shows a short. I applied ~ 1 A between the +15 V and GND pins, then traced the I*R drop along the PCB -- if there was a branch where it stopped dropping, then that wasn't the current path. Eventually I found a small 100 nF ceramic capacitor that appeared shorted -- it didn't look discolored or anything, but removing it removed the short. I soldered in a new 7815 regulator and now the 4156A powers up without any errors !