I was worried more the "unknown state" between switching ... but figured the cathode open is safe, no current ...
Smart thinking.
You have a milli-seconds moment where the cathode might be open, which is no-plate-current and can't be bad for the tube.
Unless of course you switch the cathode by shorting the cathode directly to ground and bypassing the existing cathode resistor. No plate current disruption there. The other pole of your double-pole double-throw switch will be switching the non-grid-end of your output tube grid reference resistors from ground to the negative bias supply.
Worst-possible case in that scenario is the grids are still at ground while the cathode drops to ground potential, giving 0v grid-to-cathode and max plate current. But being milli-seconds at most (and probably less time) you won't hurt the tube. You simply have the chance for popping like I mentioned, which may or may not be objectionable.
If it really worries you, move over to solid-state switching, with JFETs handling the connection for the grid reference resistors, and JFETs controlling MOSFETs for the cathode-to-ground connection (cathode resistor is always physically connected, but is simply shorted in fixed-bias mode; MOSFETs may be overkill, but you won't have to worry about current ratings when selecting them). One common switching voltage turns on/off each solid-state device as appropriate (though you may have to add extra solid-state components to invert switching polarity depending on which parts you use in each location), and switching time is likely in the nano-second range.
You don't have to worry about whether 2 mechanical parts do their work at exactly the same time because you can use a simple SPST switch to control a single voltage switching both circuits simultaneously. Then again, maybe your concerns about the mechanical switch aren't severe enough to lurch down the path of sand-state complexity. Really, it's a simple circuit in transistor-land, but takes a bunch of parts to replace a simple mechanical DPDT switch. The upside is the solid-state stuff will likely last forever, and the mechanical switch becomes much smaller, cheaper and easy to remotely control from a footswitch if you choose.