Electronic – Op Amp CMRR problem

Your calculation of the gain (assuming an ideal op-amp) is not an approximation- it is exact.

The problem is underconstrained. Try setting R1 = R3, which, of course, means that R2 = R4, from your second line above.

The CMRR will only enter into the tolerance calculation. You can make the tolerance of each resistor worst-case in the correct direction that it changes the gain either up or down (pick one).

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.