New apartment a/c issue - can't catch up

It probably needs to have the condenser cleaned out. They get packed full of lint and dirt and the unit will stop cooling. Your landlord will probably not want to do that since the whole unit will have to be removed to get at it.

I think that will eventually cause a compressor failure. Some units have a high pressure cutoff switch which will trip when the head pressure gets too high, to protect the compressor. But a thru-wall unit probably does not, so it'll keep running with excessive high head pressure, which will eventually kill the compressor.

So a landlord who knows about air conditioners will realize that (1) he can pull it to clean it out now or (2) pull it to replace it with a new one later.
We took the unit out and cleaned it today but there were no obvious obstructions. I put a meat thermometer in, and got a reading of around 54F with a room temp around 74 and exterior in the low 80s.

That’s around the normal 20 degree delta suggested by google so it seems it’s just massively undersized for the size of the apartment and the huge sun load with the big south facing windows.

I think I’ll start looking at a portable unit for the bedroom.
Looking at the picture of the window in your earlier post, it appears that those seem to be iron or steel casement windows which are notorious for having little or no insulation value. I also see by the same picture that your apartment is equipped with electric baseboard heat. That also will not do you any favors when it comes to electric bills this winter. My suggestion is that you try insulating the windows with one of those kits, similar to one below.

These are getting popular. Having the DC inverter unit will save on energy costs plus you can take it with you if you move.