Is your unit one of the ubiquitous Midea rebadges? Or at least one where the remote does 'follow me' type periodic, remote temp reporting?
If so, you're right, the easiest way is to build/buy an IR blaster module for a microcontroller (ESP32 if you want wifi) and connect a temperature sensor. HA/EspHome is overkill for this, but can be useful for the large catalog of pre built integrations. Most of them are built using/on IR libraries for 'Arduino' anyway, so you can easily roll your own with a half dozen lines of C++.
Mapping out your own PID correction algorithm (esp says the room is 72 F when it's really 70 F to get it to cool sooner), but usually just having the remote room temp sensor near where you'd be sitting instead of near the ceiling or in the return air stream gets you close enough.
Thanks for this. Yeah, it has the follow me mode that sends temperature from the remote control every 3 minutes. I think it's a Senville from Home Depot. The remote control sensor only seems to read at 1 C resolution, so it seems like using the follow me mode with the stock remote is right out of the question.
My biggest worry is the heat from the microcontroller affecting the temperature on the temperature sensor. I want to use a Sensirion SHT45, it has 0.01 degree temperature resolution and is supposed to be accurate to 0.1 C. But I need to locate the temp sensor far enough away from the microcontroller to make the overall remote controller form factor awkward. I have a Raspberry Pi 4 that does the temperature data logging and web server, but that runs pretty hot.