You could use a battery.
You get max transconductance with 1.5V on the filament. You'd have 75V on the screen (more than max, but not by too much), and I'd run the tube at 100V. With a 25K plate load, this should leave you with about the same gain as the 5670. I'd run 6mA through the tube, and use a 25K/2W resistor from plate to ground (4mA), for a total of 10mA. (90 Ohm R1)
You'll need to adjust the bias trim pot to see 7-8V on the filament so that you can get 100V on the plate of the 3S4 (I believe the "bottom" of the filament, see the datasheet), so the 4.99K resistor by the 10K trim pot should get another 4.99K resistor soldered across it to increase adjustment a bit.
Obviously, you'll need to increase the overall driver stage current to allow some current to flow through the shunt regulator. If we assume 10mA through the 3S4 and its plate load, 1mA to bias it, then maybe add 2mA for the 6J6, which leaves you at ~13ma, so a 66 Ohm R1 or thereabouts resistor will work (a little extra current can be sunk through half the 6J6 without any concern).
If you want to make 3V for the 3S4, cut the drain wire going to the "CT" on the 6.3V feed. Bridge rectify that into a 10,000uF/10V cap, then feed a 3.3V TO-220 regulator with a nice 10uF cap on the output. (3.3V is kinda high, but the battery tubes are tolerant to voltage shifts).