I'd think that if there is a capacitive coupling to the gate, one could get
a relaxation oscillation started.
It might slow the response of the opamp enough to oscillate.
I think it might depend on leakage from the source or drain to the gate.
Tinker Dwight
I removed the cap (10 nF) from the gate to the analog feedback line and left the cap from the output of the op amp to the feedback line as well as the resistor (which the OP removed).
I'll have more information tonight when I scope it out.
BTW, maybe nobody noticed but if you right click on the photo of the meter and select "Open image in new tab " you'll see the P.S. used to power the op amp is set to 8.5 V instead of 12V or 10.5 V. This is because I noticed that the voltage across the resistor actually INCREASED when I lowered the op amp power voltage from 10V to 8.5 V. It went from 4.88 V to 4.96 V.