I did this with ESPHome on an ESP8266 connected to an IR led for some rope lights that had IR control. The hard part is finding the codes for each control, but there's a surprising amount of info out there.

Even if you build your own thing to communicate with the AC, Home Assistant is still useful since it lets you easily automate things and interact with other devices, and you get a bunch of things included (nice UI, storage of historical data, dashboards, etc). You could build your thing as a Home Assistant integration.

That might depends on how your AC is working, I recently did the same at my dad's house and run into issues because the temp control for the unit is reading from the remote. So when I sent infrared command to turn off unit, it worked as expected for some minutes then turned back on because remote sent back command as temp wasn't meet. Still need to do a lot of testing to find a way to make it work, might need to disable remote and use some temp sensors for the triggering part but not sure how that will work, might need to use some hysterisis friendly climate addon for that...