I solved my problem after poking around for most of the day today!
With Paul's guidance (thank you, Paul!) on ground-related issues, I took a more methodical approach today when I noticed something very strange. The noise profile I was seeing was present
even with the amp turned off! Turns out, just having it plugged in was causing all of the noise I was seeing in the FFT graph see first image - "Power Off- Plugged In- No Cheater"
With Paul's hint of a grounding issue, I pulled the plug from the wall and inserted a "cheater plug" that has no ground lug on it, and plugged it back in. The amp was still off, but the noise floor dropped all of the way down to -115dB (except a little peak at 60Hz). See the second image "Power Off - Plugged In - Cheater Plug"
Since I wasn't really excited to run an amp with a 450v power supply without a ground connection, I tried something else. I inserted a CL-60 thermistor between the AC Ground/Chassis Ground and the PSU/signal ground. So the aluminum plate is still directly bonded to mains earth, but the rest of the audio ground is separated by the thermistor. It measures about 15R when cold and can handle more than 5A of current, so I figure it's plenty robust for this application, especially since the amp itself is protected by a 1.25A fuse. This way, any potential fault will still find its way to mains earth for safety, but the signal ground is a bit isolated.
So, the final image here is my noise floor with the amp powered up with the CL-60 thermistor in place. See "Power On - With CL60."
Playing with the hum pot impacts the 120Hz and 240Hz peaks. By adjusting the pot, I was able to reduce the 120Hz peak from -66dB down to -80dB. At the same time, the 240Hz peak went from -83dB down to -92dB. This is easy to see in the FFT, though is just *barely* audible through headphones. The amp is now *VERY* quiet. The overall AC hum at the speaker terminals is now about 0.65mVAC - less than half of what it was before! With a little more tweaking tomorrow, I think I can get these two peaks down just a bit more. I can't seem to do anything with the 60Hz peak, though. It pops up as soon as the switch is turned on.
I am just thrilled with this amp now! Now that I've solved this issue, it's time to rebuild the second amp on my power coated aluminum base.