There is a bit of tolerance on the meter and what actual current will show, that's not super surprising.
When you change to the 930 ohm resistor, the shield pin on the 5670 goes to the same grounding point as the signal ground to the chassis, so I wouldn't be too bothered if you just put the 930 ohm resistor between the shield pin and the driver cathode pin. You can leave all the components on the PC board related to biasing up the driver cathode and just disconnect the wire between the PC board and driver cathode.
Very technically, running the cathode bias resistor between the shield and cathode is "wrong", but in this instance the signal levels are large.
To convert to a #45 tube, you need to drop the B+ a lot (as you've observed), the 0.1 ohm filament dropping resistor needs to be increased since the #45 doesn't draw as much filament current as the 2A3, and lastly you'd adjust the cathode bias resistor. The 3K output iron can work with a #45, but it will require a little bit of a different operating point compared to what's in the Stereomour. I would want to run the #45 at the maximum allowable current, but a cursory glance at the datasheets isn't exactly informative as to what that is. Maybe 250V PK at 40mA would be a place to start (or buying the SEX iron ugprade and using the 4K transformer out of there instead of the 3K transformer in the MonAmour).