Possibly not something that exactly matches your wishes, but I would suggest that you look for a TM-902C thermocouple thermometer. They are very low cost and I would be surprised if they are not available anywhere in the world. A couple of potential advantages: firstly you will probably get better initial accuracy (even if possible to modify your multimeter, you would probably need select-on-test resistor(s) for adjustment, involving multiple disassembly). Secondly, you get to use your multimeter for measurements and monitor temperature at the same time.
You will probably find the TM-902C not that much more expensive than buying just a thermocouple probe.
P.S. If you go in this direction, get one with a simple slide on/off switch rather than a push-on button. Battery life will be considerably better.