Two options have occurred to me: ...
Option 3: Place the amp in Standby, then switch, then place the amp in Play.
I'm no switching expert, but one thing occurs to me: If you make a change that causes a d.c. shift, there will be a thump no matter what. Maybe not in this case, because there is a filter cap that could soften the transient.
Okay, okay... the real Option 3: Whatever your switched parallel resistor is, place a real big resistor in series. Say, 220k-470k. Place your switch across that big resistor so that when the switch is closed is shorts out that resistor. Open, you have a big honkin' resistor in parallel with your existing dropping resistor, so almost zero resistance change. Closed, your added resistor is in parallel (big resistor shorted) and the total resistance is reduced as expected.
About the desired total resistance value (for changed screen voltage):
Simple rule is that kilo- means 1,000 and milli- means 1/1,000. This can be used to make mental math easy. If you wanted a 25v drop and your screen plus preamp draws 25mA, you could use a 1k dropping resistor. If your screen plus preamp really only draws 2.5mA, you need a 10k dropping resistor to get a 25v drop. Consider this to zero in on your resistor value.
And one question. What exactly are you trying to do, and what do you hope to achieve with the change? I'm guessing you want to reduce the screen voltage while in fixed-bias mode to compensate for any difference in screen/plate-to-cathode voltage reduction when you're in cathode bias mode (or maybe you want higher screen voltage in cathode bias mode).
I wonder (and don't know for sure without experimentation) that a 25v screen voltage change will make an audible difference. At least, I don't know that you could hear the effect when it is steady and not due to screen voltage compression as the tube tries to make maximum power.