Regarding swapping a 12AU7 for a 12AT7—you’re correct in that the 12AU7 has less gain and lower current draw. Since you’re using a 2.0V RMS source, the gain reduction may not be a significant issue, and it could still work fairly well with the reduction in overall gain.
The 12AU7 has about 1/3 of the gain of the 12AT7, so you’ll get less overall amplification. However, because you're using a 2.0V RMS source, this may still be sufficient for your needs. You may not need to adjust the current source (CCS) or cathode resistor (365 ohms) as they are designed to work for a wide range of tube variances, and the lower current draw of the 12AU7 likely won’t require significant changes.
However, the reduced current draw of the 12AU7 means it would likely draw less current from the power supply, so the bias point could shift slightly. You can try it as is, and if you notice any unwanted distortion or tonal changes, that might indicate the need for small adjustments to the CCS or cathode resistor.
It will likely work but with reduced voltage gain, so the overall output might be lower, but it shouldn't hurt anything in the circuit.
If you feel the output is too low, adjusting the CCS or cathode resistor could compensate for that loss, but if you're just looking for a quick test, trying it as is could give you an idea of how the circuit behaves with the 12AU7.