Patrick, Never tried it, but my understanding is one simply needs to tie the screen and control grids directly together on the 4-400 to make it a zero bias Class B triode. This is what is done with 813s. Not sure what kind of a peak grid-to-grid input voltage swing would be needed but 1 KV seems kinda high. Have used 807s with the 20K resistor setup. Worked FB. Have also had good luck with sweep tubes by grounding the control grids and driving the screens only.
Give your scheme a try and let the rest of us know how it works out. Dennis D. W7QHO Glendale, CA