I'm renting at the moment and the AC units are the non-inverter type, i.e. the compressor is either going full power or off. When we run them the room gets really cold when the compressor is on and then the compressor stops for a few minutes and it gets unpleasantly warm and sticky again. Can these non-inverter units be adjusted internally so they keep the temperature in a narrower range? i.e. compressor switches on and off more frequently.

Brand is Seijo Denki, Superior

I doubt an affordable technical solution, perhaps have a look at why the temp is climbing so rapidly and try to slow that down by other means?

Shade windows (ideally from outside), weatherstripping, wall-hangings, rugs. . .

Also try geting your body more acclimatized, keep the temp higher all the time - I've lived here fifteen years, last 13+ with no A/C at all except for my customers on the first floor, only on when they're there.