I once again pulled out my troubled 3456A to try and fix it. This time I succeeded at finding the issue. U303, an LM201 configured as a non-inverting amp to produce +30V to power the input FET stage. It was only outputting about 4V, so I connected an external supply to provided the 30V, and it works perfectly, all error codes gone and minimal current draw. Checking the inputs and outputs, it's clear that U303 is dead.
So I can easily get hold of LM201A in a plastic DIP, but the originals are metal can TO-99 package.
It seems like a fairly benign part of the circuit, so is there any reason why I wouldn't just drop in a plastic DIP replacement?