I've revisited this repair over the past few days. Today, I fixed the 5V power supply problem with a new bridge rectifier. I drilled through the carbonized area of the PCB and soldered a bodge wire directly from the positive leg of the rectifier to the corresponding capacitor. The unit powers and goes through the self test.
Now, I have some weird problems with the output. If I set negative polarity, I can get output on all the ranges for all possible values most of the time. (Sometimes I get an overload when I am increasing the voltage) If I set positive polarity, I can only get output up to about 0.5 V on the 100 V range or about 0.2 V on the 10 V range. Anything else would trigger an overload, and I would continue to get overload until I back the set value down to near 0. At first, I thought this could be a front panel switch issue, so I sprayed each switch out with deoxit and actuated them many times. This does not seem to solve the root problem, however. I get similar results if I attempt to program the output over GPIB, but for some reason I can only reach 0.2 V on the 100 V range when setting remotely as opposed to the 0.5 V I can obtain when setting locally.
Are there any suggestions for how I should proceed to troubleshoot this?