DON'T use your scope hps50 to check the ripple of the power supply
It only accepts a peak voltage DC+AC of 100 V, and the power supply in the RF generator has for sure more than 100 V DC.
To just put a resistor is always tricky. I have seen 39 years ago, how a coworker destroyed the vertical attenuator of a Tektronix 465. It only accepts 250 V and he tried to measure 300 V AC using a x10 probe in a switching power supply.
The 400 Hz are generated using the 6AR5 as a Colpitts oscillator, this requires the use of a large inductance and has no automatic level control, so don't expect a nice waveform. Mi IG-42 uses the same method and the 400 HZ don't look nice either. When I want a good looking modulation, I use an external audio generator.
You can measure using a DMM the DC voltage at the output of the power supply and then switch it to AC and measure the AC component. Even if your DMM is not true RMS, you will have some idea of the ripple. If it is comparable to the amplitude of the bouncing that you are seeing, probably the problem is the power supply. If this is not the case, the 12BH7 "could" not be completely OK and make the oscillator work unstable.
Tubes are quite resilient. To damage the new ones just by using them some minutes for testing, would require more power that the one that the power supply of this generator can usually deliver.
The clipping is normal in this kind of tubed generator, with grid current. The amplitude of an oscillator depends of the loop gain at the oscillating frequency being " 1 " (Barkhausen criteria). Without AGC, this involves some clipping
To get a perfect waveform, you need a much more elaborated generator, like the HP-606B