How original do you want to retain the power supply circuitry?
For example, how about replacing the 6BH6 with a MOSFET/transistor?
This would free up the filament winding to use for your external vacuum tube device.
If you still wanted to "light up" the 6BH6, you can leave the plate, cathode and grids disconnected but connect the filament in parallel to either the 6L6 filament, or your newly ground referenced filament for your external vacuum tube device.