In actuality, the first Crack that Doc B. and I built ran the 12AT7, but it had way, way too much gain for any headphones that we tried. The 12AX7 will be much worse.
A 5687, 12BH7, 6CG7, or 12AU7 are the families of tubes that run well in this arrangement, with the 6CG7 having a different heater arrangement and the 5687 having an unusual and very different pinout.