Something here definitely stinks. If its a year old, it’s under manufacturers warranty at the least for parts if there’s actual holes in the coil. But a cool shouldn’t be getting so filthy it needs a chemical cleaning in one year of use even without a filter unless you’re running a pet grooming shop in your house. I’m not sure it’s your landlord trying to screw you so much as your HVAC company trying to screw your landlord.
Either way, you definitely need a second opinion, and your LL should pay for it. Make sure it comes from a reputable company that’s a factory authorized dealer of a reputable brand of HVAC equipment and check their BBB ratings.