How do I keep this VCCS stable?
There's a pinpoint comment which brings up the key concept: the stability. You have designed a feedback loop which tries to push AC as a constant current source. Capacitive load causes immediately 90 degrees phase lag and that can be enough to make your loop an oscillator when added to already existing phase lag.

Your options:


reduce the gain at high frequencies DIFFICULT! Causes easily more phase lag.
insert a permanent minimum load resistor and take it into the account in the calculations. That resistor hopefully reduces the phase lag caused by the capacitance.
redesign the regulation to work in DC mode ie. the current is forced to have amplitude which causes the right peak voltage to the current sense resistor


ADD: The case is extremely complex. Capacitive load affects more at the lowermost end of your current sense resistor, but if we assume the differential amplifier function still be valid at high frequencies, the current sense voltage is actually RC high pass filtered. There's a slight possiblity to compensate the case by adding a small capacitor in parallel with the current sense resistor. No quarantee, but easy to test.

More complexity is caused by the fact that wirings have also inductance. There can be parasitic resonance circuits at megaherzes or tens of megaherzes. DC mode regulation would solve HF phase problems.