That is not how that works at all. For starters only dogs and miniature horses are even recognized as valid protected service animals. And there is no valid badge or certification for that.

While emotional support animals can have a protected class for housing in some circumstances it would at a minimum require a recognized medical disability and documentation from a medical professional confirming your need for that animal medically.

Buying some worthless patch off Amazon is going to hold absolutely no weight whatsoever legally. Even after all of that you would most likely just be asked to move there is a reason that certain places are not pet friendly and it’s to give people with allergies and other issues with pets a space