Optimizing op-amp overload recovery behaviour

Put 3 low-leakage low-capacitance diodes in series. Now, stick that in parallel with your feedback resistor R9.

When the opamp's output voltage becomes high enough, the diodes will conduct, and your 200x gain turns into a unity gain follower.

This will prevent the opamp from clipping, so it won't have to recover from clipping. It will not turn this slow opamp into a fast one though. You can always try, it is simple and cheap.

Am I doing something horribly wrong

The op-amp chosen is wholly unsuited to your expectations. The gain bandwidth product is 2 MHz and that means, as a unity gain amplifier, it will have a bandwidth of 2 MHz. With a gain of ten is will have a bandwidth of 200 kHz. Withy a gain of 100 its BW will be 20 kHz. You expect a gain of 300!

Also, the amplifier step response settling time is 150 us typically.

If not, can I get faster output response from this setup

To operate the way you plan, you need a much, much faster op-amp or you need to find a way of clamping the heater voltage to virtually zero volts so that the op-amp isn't having to recover from saturation and deal with a massive step response.

Is there a more appropriate approach to doing this (in hardware perspective)

I would consider using a sample and hold technique so that the thermocouple is disconnected from the input just prior to the heater powered on and reconnected just afterward the heater is powered off. Use a 100 pF capacitor for "holding" the thermocouple voltage when not connected.