Most thermocouple probes for high heat are of the K type and work fine in the interior of your RMH.
A meter for reading K-type probes is readily available at Amazon.
https://www.amazon.com/Proster-Thermocouple-Thermometer-Dual-Channel-Thermocouples/dp/B071V7T6TZ/ref=sr_1_4?sr=8-4
These meters all claim to be compatible with various types of probes, including the N-type.
It has been brought to my attention that this is not entirely true.
A meter for a K-type probe will not give accurate readings for an N-type probe.
Unfortunately, I have discovered that the K-type ceramic high-temperature probes have a problem performing in an oxygen-starved environment.
The K-type probe inside Shorty's riser started failing to read within just a few weeks.
Glenn Littman in Colorado has had the same problem with his K-type probe inside the riser.
Glenn contacted Carston at Minnesota Measurement
https://www.meter-depot.com/
He gave us the following information.
In an oxygen-starred environment, a K-type probe will develop "green rot" and stop sending accurate signals to the meter.
An N-type probe has a similar cost to the K-type but can last for many years in a low-oxygen environment.
He informed us that the K-type meter has a different calibration curve and will not read accurately with an N-type probe.
The different alloys used in construction will result in different voltage outputs for the same temperature.
Minnesota instrument's will be carrying an N-Type meter soon.
A quick search on Amazon failed to locate any meters calibrated for strictly N-type probes.
Despite the claims, the low-priced K-type thermocouples are only useful for reading internal bell temperature and exhaust chimney gas temperature.
Experimental Rocket Scientists who want this information should invest in all N-type temperature probes and a matching N-type meter to obtain the proper readings.