MarkT:
If you have a tachometer you can measure the rotation speed for a fixed
supply voltage directly, which you can then convert to volts-per-radian-per-second.

This then tells you the newton-metres-per-amp directly, they are the same
constant.

Question for you... I had the impression that the voltage in the motor constant, when expressed as V.s/rad, referred only to the back-EMF. So when the motor is at peak no-load speed with constant current:

Vterm = I*R + Vemf

the number that you care about is really Vemf, which is the terminal voltage less the motor current times the winding resistance.

Consider two geometrically & magnetically identical motors, one wound in Al and the other wound in Cu. The former will have higher winding resistance, but they will have electromagnetically identical behaviour (i.e. same current/torque relationship) because the current takes the same path. If you measure the motor-constant using current & torque, they will both give the same answer. To get the same answer with voltage implies that you are measuring the Vemf and not the terminal voltage.

Or am I missing something there?

Anyway, with your typical cheap crap little DC motor, the voltage lost to winding resistance can be significant, so if you want to measure motor-constant, you need to be aware of that. You must measure all of terminal voltage, current, winding resistance and speed in order to arrive at a motor constant:

MC = omega * (Vterm - I * R)
omega = RPM * 2 * pi / 60