• WARNING: Tube/Valve amplifiers use potentially LETHAL HIGH VOLTAGES.
    Building, troubleshooting and testing of these amplifiers should only be
    performed by someone who is thoroughly familiar with
    the safety precautions around high voltages.

To bypass or not to bypass...

The screen grid needs either to be stable (pentode mode) or have a proportion of the anode signal (ultralinear mode). You are somewhere between the two, as the screen current will roughly follow the anode current (except at negative peaks) and so will develop a signal voltage across the resistor.

If you have an oscilloscope look at the screen voltage. If it is a good copy of the signal then bypassing or not is up to you. If it is distorted (look at negative peaks) then you need to decouple or use a smaller resistor value.
I have an old HH Scott model 299C amplifier. All of the screens on the 7591A output tubes are tied together and fed through a 1.2k resistor from the B+ supply with a 15k resistor to ground. I have been concerned about this arrangement for a long time but since the amplifier sounds great, I haven't added a shunt capacitor on the screens. Perhaps I'll try it to see if there is an improvement in perceived sound quality.
Does anyone have an opinion about this?
When a P-P pair share a screen resistor then there is cancellation of the signal and odd harmonics at the screen, although even harmonics add and can recreate odd harmonics by second-order mixing with the original signal. This is for smaller signals, where the screen current roughly follows the anode current.

When the anode voltage dips below the screen voltage (on negative peaks) then the screen current increases relatively faster than the anode current so there is extra distortion. In a P-P pair with common screen resistor this will happen on both peaks. It might help smooth off peak limiting, or it might make things worse. Try it and see? If possible, use an oscilloscope to look at the waveforms.
In most power pentodes, the sum of the plate and the grid current is almost constant.
When the anode current suddently drops because the anode voltage is going too low, then the screen current "sky rockets".
This is dangerous for the tube, specially if a bypass cap tends to keep the screen voltage hi.
Unbypassing is safer althought this indicates that the amp is hardly overloaded f.e. because the output impedance is too high.
At this point, the gain miserably drops and any NFB loop make the things even worst.
The only way is to turn the volume button down !
However, if as a guitar player you like this overload effect, use unbypassed serie resistors in each screen much larger than indicated in the data sheet . . . or have a box of spare tubes at hand.;)

You can check in various DataSheets that -for a given grid and screen voltage- the screen current decreases when the anode voltage and current both increase ... slowly !
But below the knee, the screen current rises as suddently as the plate current falls and then the screen dissipation is largely exceeded ... momentarily !

Predicting the "right" value for a serie screen resistor can only be done if all the other parameters (B+ voltage, Bias, Load and Tube's limits to name a few) are already known. In doubt use a larger one :cool:

As already said, one parameter is the "duty factor": an instrument or public address amplifier is more likely subject to be frequently overloaded than an hifi rig ;)

I had in mind the fact that output valves are usually driven from the control grid. For a given screen voltage, as the control grid voltage changes the anode and screen currents are roughly proportional until the anode voltage dips below the screen voltage. Data sheets show this.

When the anode voltage is significantly above the screen then the current division is largely decided by geometry. When below the screen the screen grabs most of the cathode current, as you say.