At least for some tubes, the solution is to connect a (very) high voltage diode between g2 and anode, diode cathode to tube anode (reverse biased). Should be done near the tube, but outside the RF compartments, of course. This way the screen supply also feeds the anode, if the anode supply is disconnected/not yet ready. The diode should be able to carry the full anode current, and be very generously specified. A failed diode would not be too much fun (but should be caught by the tetrode board).
You need to check whether this keeps the screen current sufficiently low to not cause any problems (running anode at g2 voltage), or if the total anode+screen current is too high for the screen PSU. In a good beam tetrode the total cathode current should stay the same if the screen voltage remains unchanged. So the anode 'fault' current is likely to stay around the same value as seen during normal operation, causing a significant g2 PSU current drain when the anode supply is absent.
Second option is an electronic interlock, which cuts the tube current to zero by reducing g1 bias voltage anytime the anode supply may be absent. Your PSU should already be able to do this. However the reason this option is second in line, is that the diode protection is much simpler and thus less likely to fail when you really need it. You may want to use both techniques. For everyday operation the g1 bias cuts the tube off as and when needed. Yet should this somehow fail, the diode routes the g2 current to the anode, possibly causing your Tetrode control board to trip on high g2 current, which still saves the day (and the tube).