Disclosure: I haven’t done ac work in 20 years, I never really worked on any ptac units, and you should know what you are doing before you mess with any electrical work. This information is general and I have no idea what the condition of your ac wiring or home wiring is in, so don’t follow any of this advice without researching and understanding what you are doing. Safety is more important than comfort. These are just ideas for you to look into.

A thermostat is a fancy set of switches.

Red brings the power in and internal switches turn on the heating, cooling, and fan cycles depending on settings. If everything is working except cooling, you may need to install a jumper between the R and the Rc on the thermostat or look into internal settings of the thermostat to get power to your cooling cycle. Sometimes units have separate power supplies for heating and cooling cycles due to different methods of heating. If everything is electric, it probably needs to be jumpered to supply power to the cooling cycle, but that can vary depending on model.

You can verify power not getting to the cooling cycle by checking voltage between the R and Y connections on the thermostat. You should show 12v between R and Y on the tstat if the cooling cycle is on and no cold is coming out. If you read 0v and the fan is running, the thermostat is working fine and there is a setting on the unit that needs to be bypassed. If you are reading 12v, you can try using a jumper wire to connect the R and Rc to see if the cooling cycle turns on, but I wouldn’t start jumpering things without knowing what voltages are present. If you don’t have a voltmeter, wait for advice from someone that has experience with those units.

There is also a bypass procedure to control those units from a remote thermostat that takes the local thermostat on the unit out of the circuit

https://m.youtube.com/watch?v=4uFp58yHd64

This YouTube video shows the process.

Be safe and don’t do anything you aren’t comfortable with. Please don’t hurt yourself.

https://hvacdirect.com/media/hvac/pdf/AMANA-PTAC-Service.pdf

https://digitalassets.resideo.com/damroot/Original/10003/69-1775.pdf

Those are the ones you have, right? Check the wiring for each function to make sure the thermostat and unit agree. Then check the configuration on the unit and thermostat to make sure it’s correct. Then run a system test from the thermostat.

You should also be able to just short Y to R to turn on the compressor.

But I’m not a professional, and I’ve never worked with anything more than a furnace with a fan. But it looks like the principle is the same.