You will probably need to increase your voltage. I haven’t ever used the LM334, but it will need a minimum voltage across it. I don’t know if you are still using the IRFP250N, but if so it has a threshold somewhere between 2V and 4V for 250uA, so the threshold won’t be as high but it should be close. I would try using 5V if it’s fine with your setup.
For the suitability of the resistor method, you should do the math on how a change in Vds will affect the current, and then calculate how much this variability in current will affect your readings. If this error/innacuracy is acceptable, then why not.