They'll call your doctor and go to the full extent of the law if your animal is really an ESA. Sure you can still go to a doctor and mention that you're depressed an how an animal will really help you out...etc.
But I wouldn't assume that large apartment complexes just stand by idle. Most are tired of fake ESA animals that cause them insurance & tenant issues.
You may be suggesting that lying about service animals is easily discoverable, which is indeed true. This is already illegal.