Overshoot on MOSFET drain
+1 for next-hack.

TL082 is a slow opamp with anemic output current. It's perfect for some other uses, but here... 3MHz GBW, 13V/µs slew rate. You shouldn't expect something with a 3MHz bandwidth to do anything useful in the 150ns time window you mention... besides, its output current is tiny, it is unable to drive a 100 ohm load, so don't count on it to push current into the capacitive gate of the FET with any enthusiasm...

Now you say this is a function generator, but it has variable output impedance: when the bottom FET is ON, then its output impedance is close to 0R. When it is off, then it is the unspecified and frequency-dependent output impedance of the bottom FET drain and the top FET controlled by the opamp.

If you intend to drive a 50R cable with this you might have problems.

Now, you got several options:

The opamp + MOSFET is a voltage regulator, and usually at the output of these, capacitors are added to clean up the transient response, because the feedback loop is not infinitely fast. Note that even with a faster opamp, you will need to carefully work on frequency compensation to make it rock stable.

Or you can kill the planet: Use a differential pair as a switch. Or two MOSFETs with complimentary drive. This throws away the current into ground when not in use, and sends it into the load when it is in use. Thus the amount of current drawn from the reference voltage is always the same, and it will have no settling issues.

Also note that you could use a smaller, faster FET to reduce capacitive shoot-through.