For now I have driven the C-field solenoid using my Fluke 732A and a Vishay Z201 2.5k resistor.
But is that good enough ?
In this test I put a very light piece of plastic foam over the physics package which raises the outside temperature roughly 3°C.
That makes the current through the C-field coil drop 72 nA
A C-field shift of 72nA will cause a frequency shift of approx 3.4e-14.
For an active C-field driver to do it better, it must have total noise, all causes, less than 6 PPM, which isn’t trivial.
72nA change on top of 4mA at 10V is a change in total resistance of 0.046 Ohm.
The resistance of the C-field solenoid is approximately 12 Ohms, so that is approx 0.046 / 12 = 3833 PPM
Copper has horrid tempco, around 3900 PPM/K, so it is pretty good guess that the solenoid got approx one degree warmer.