Phase splitters/inverters using valves with directly-heated cathodes
Obviously, a directly-heated cathode means that the heater power and the cathode signal share the same pins. But this does not preclude the circuit configurations you're considering, it just means that you need to decouple the signal from the heater power somehow.

This means that the heater power needs to "not care" about the signal voltage, and vice-versa. Since the heater power is going to be DC and your audio signal is AC, there is a number of ways that this can be approached.

If you want to use all passive components, you need a low-pass filter for the DC and a high-pass filter for the AC. However, if you want a relatively low cutoff frequency (presumably a few 10s or 100s of Hz), the components can get inconveniently large.

If you're willing to entertain the idea of additional active components &mdash; perhaps even transistors &mdash; you can control the heater power by means of a current source. This will have an inherently large effective source impedance, which will allow the signal voltage to be tapped off with no interference.

The filament only requires 10 mA @ 1.25 V, so either approach is reasonably feasible.