I am impressed by the circuit you have designed. Here are some comments on it (I'm an electronics nut. Please don't interpret anything as criticism, I just like to explore different ways of doing things):
The differentiator capacitor in your circuit does not need to be a bipolar type - it can be an ordinary polarised electrolytic with the negative terminal connected to the op amp input - provided that the temperature signal from the thermocouple is always positive, and with roasting I would expect that to be the case.
Here is my reasoning, but you can verify by measuring the voltage on the capacitor in your unit using a DMM (measure both when the temperature is rising and when it is falling to be fully convinced):
The inverting input of the op amp, to which the negative terminal of the capacitor should be connected, is a virtual earth, since the non-inverting input is connected to earth and there is negative feedback. The voltage at the other end of the capacitor is always positive if the temperature signal is positive. In fact, the voltage across the capacitor is exactly the temperature signal (it only deviates from this at power-up when the capacitor is initially discharged and the op amp output will swing to the negative rail in a bid to bring the inverting input of the op amp back to earth potential, an action which results in the capacitor charging fairly quickly to the temperature signal voltage).
Your design needs the op amp to be offset-nulled because the meaningful outputs are of the same order (a few millivolts) as the op amp's offset error. You can reduce this problem by a factor of 10 (or more - subject to having sufficient output voltage swing from the op amp, and also subject to the differentiator resistor and capacitor not becoming too large) by having the differentiator output 10mV per degree per minute (rather than 1mV/degree/min) then use a potential divider to divide the op amp output by 10 so that the DMM can still display the rate-of-change directly in mV/degree/min.
Of course, this probably means reverting to a 1000uF differentiator capacitor, so the start-up problem rears its ugly head again. I have designed a solution for that - years ago I designed a temperature rate of change meter for environmental temperatures, which required a 0.047F supercapacitor in the differentiator, and the start-up problem was so severe that I incorporated circuitry to force extra current into the capacitor, bypassing the differentiating resistor, when the op amp output went near the rails. The extra circuitry involved a handful of cheap transistors, diodes, and resistors (I hope you have large hands, though

)
Another trick to reduce the start-up problem is to offset the temperature signal (by subtracting a fixed voltage from it) so that it is zero at the lowest temperature of interest. This reduces the amount that the differentiator capacitor has to charge at power-up.
Also, subject to still having enough voltage swing available from the reduced supply voltage, you may be able to power the whole thing off a single 9V battery without the inverter i.c. if you use another op amp to generate the earth rail. Connect a pair of 100k resistors across the 9V supply, wire an op amp as a voltage follower (i.e. a non-inverting buffer), and connect the buffer input to the junction of the two resistors - the op amp output becomes your ground (0V) potential, the battery positive is then +4.5V and battery negative is -4.5V.
The final blow-everything-out-of-the-water-and-start-from-scratch idea is to go digital. I have designed a temperature rate-of-change meter based on a PIC 16F628A microcontroller, a 16x2 LCD, a 32kHz watch crystal, and a DS18B20 temperature sensor that does everything you need (it even displays the temperature!)
except that the DS18B20 only works up to 125 degrees C. If there is enough interest from this forum I might modify it for a thermocouple input.