Op Amp with positive feedback (hysteresis) driving N-MOSFET oscillates
After a reading your question and having a quick look at the datasheets of the NCS333SN, I am sure that the problem is the heavy capacitive loading of the amplifier by the DMN1019. The two details motivate my belief


The "Dynamic performance" parameter table at p. 6 states in each entry (apart from the slew rate SR entry) a value \$C_L=100\mathrm{pF}\$ for the load capacitance. This suggests that it is not advisable to increase too much beyond that limit the load capacitance.
When you have a look at the "Typical characteristics" section, looking at figure 1, p. 7, you notice that the nice phase margin at \$f=f_T\$ of the amplifier is a more than respectable \$\phi_M\simeq 60^\circ\$  but again when the load capacitance is \$C_L=100\mathrm{pF}\$. And if look at figure 13, p. 9, you see that the small signal overshoot is over \$60\%\$ when  \$C_L=1000\mathrm{pF}\$.


Since the typical gate capacitance of the DMN1019 is \$C_\mathrm{G}&gt;2500\mathrm{pF}\$, we'll surely find troubles if we connect the OpAmp output directly to it: and even using a series gate resistor may not get you out of troubles, if this resistor is significantly lower respect to e.g. the standard load stated for the slew rate test, i.e. \$R_L=10\mathrm{k\Omega}\$, as you have noticed with your tests.

What could you do?


Since you use the MOSFET as a means for discharging a supercap when the voltage across it starts to rise above the safe \$2.5\mathrm{V}\$ level, you do not need to be particularly fast in turning it on. Therefore you could try to put a \$10\mathrm{k\Omega}\$ gate resistor and see if the amplifier remains stable.
Otherwise, if you desire to have nevertheless a quick response, you should try to find an optimal value for the gate resistor \$R_\mathrm{G}\$, by starting from \$R_\mathrm{G}=1000\Omega\$ and rising it until the circuit is stable for all desired resistor loads.


Final note

Following mkeith's comment above, I think is a very good idea to find a low voltage comparator and use it instead of the NCS333: despite their circuit topology can be (also very) similar, OpAmps and comparators cannot be used interchangeably without any care. Just to give some examples, devices like TLV3691 or NCS2200 could be a nice choice.