Interesting - thanks so much for posting!
My best guess at this point is that it's common-mode noise. Capacitors aren't going to help unless they have lower impedance at 60Hz than the two 100 ohm resistors, which are effectively in parallel. 50 ohms at 60Hz is 53uF, that's getting pretty silly.
You can replace the resistors with a pot, probably anything from 50 to 500 ohms will do. You'll want a 3-watt or greater rating. And make sure the tubes are the same make - some have series heaters, so the best balance will be off center for those. (Also make sure they are the same phase - pin 7 to pin 7, 8 to 8.) Adjust it for the lowest hum.
The Schottky diodes don't have reverse recovery spikes, but they do still have switching noise. So it's possible that a common-mode choke will have an effect. If you put one between the transformer and the rectifiers, it will reduce coupling to the other windings (i.e. the HV winding). Putting one between the DC filter and the tubes will have more effect on what noise gets to the heaters. I have, at this point, no idea which would be more effective, or whether either would be audible at all. But it's a standard approach, so there's a good chance.
Before getting too carried away, do consider getting a second set of tubes. I imagine that different tubes, especially different makes, might have different sensitivity to heater noise coupling.
