From my measurements it seems that the bias supply is stable. Without and with power tubes the AC voltage has only 0.5V fluctuation over a 10-15 min period. DC voltage about 1 volt of fluctuation after the diode and bias junction. But that is on a 25mA difference on the power tubes!
1V can't cause that much current increase. You've got something else happening.
Would it be safe to say we can rule out the transformer (feels cold as well) and bias circuit?
The output transformer temp check is a "poor man's test" for ultrasonic oscillations. Sometimes, if an amp is oscillating at high enough frequency and amplitude, the OPT will heat up pretty quickly.
Tubetech mentioned poor lead dress in the NFB connection causing an oscillation (good call!). I'd thought about this next thing earlier but didn't say anything, because from your description the tubes were already hotplating while just playing the amp normally. For future reference, I've had a few Marshalls oscillate when the power tubes are plugged into bias probe sockets. If you disconnect the NFB wire from the OPT to the PI it generally stops, so I just preemptively unsolder it before biasing new tubes.
I did notice that one of your PI coupling cap to grid stopper signal wires is running parallel to the main B+ ground. That can cause issues, as there are heavy currents flowing in that ground wire. It might be good to float the signal wire up in the air, or at least away from the ground wire and chassis in that area.
An example of what improper lead dress can do: I once worked on a hi-fi amp that was rewired with silver signal wire, and it was oscillating afterward. This particular amp was poorly designed in many ways, but the physical layout was especially horrendous. Among other things, there were signal wires routed
directly against one of the output transformers, which were inside the chassis. The OPT's magnetic field was inducing a voltage in the silver signal wires, which caused the oscillation. Silver has 200% higher magnetic susceptibility than copper, so the same run in copper didn't cause any major issues. I re-routed the wires as best I could, which stopped the oscillation.
Another example: some blackface and silverface Fenders make an annoying "ticking" sound when using the tremolo circuit, which is easily solved by tidying up the lead dress, running a direct ground, and installing a .01uF cap. AAMOF, I have a friend's silverface Twin on my bench right now, that was ticking before I fixed it 8 or 10 years ago. I bet CJ, Tubetech and some of these other guys have fixed a thousand of 'em.