As you know, ESA's do NOT have access while Service Dogs do. ESA's are not service animals. As you note.. Butters is a pet.
Some airlines allow ESA's to ride in the cabin. Others do not. Many are going to NOT allowing ESA's to fly in the cabin. They are not trained service animals and do not have access.
When flying, the safety of ALL the passengers is the highest priority. If an ESA has an issue, it is not like they can stop and "let you off" a the next airport (though if the disruption is serious enough they will and I suspect the individual responsible for the incident may get a bill!).
While Butters may be perfectly trained and lie at your feet (or wherever he/she would ride) the entire flight and most would never know he/she was there, many ESA's are not well trained and behave like pets. They do not perform a service to their handler that the handler cannot do without the dog (such as guide dog work or seizure alert).
It is my opinion that ESA's should not fly in the cabin (unless the airline allows pets to ride in the cabin), they should not be allowed in stores (unless the store allows pets) and they should not be allowed in Restaurants (unless pets are allowed). Letter or not, ESA's do not have access and are not service animals. Butters should ride in the hold like any other non servicedog (FWIW competition dogs ride in the hold). Your better bet might be to board Butters or hire a pet sitter and leave Butters home.