This might be of interest: Capital punishment by the United States military - Wikipedia
Eisenhower and then JFK approved the last actual execution of a U.S. serviceman: John A. Bennett - Wikipedia
Charcoal burning would be my preferred method, probably combined with anaesthesia of some sort.
A few years ago, Michael Portillo looked into various methods for a Horizon documentary and settled on hypoxia using some more-or-less inert gas (most likely nitrogen) based on veterinary studies (and some time in a hypobaric chamber).
Interestingly, the notion wasn’t all that well-received. The folks he spoke with at the end of the program felt it was too humane.
For thoroughly gruesome details, go to Hanged by the neck until dead. The process of judicial hanging. Actually, after the introduction of the UK drop tables in the 19th century, if one must have capital punishment, then long-drop hanging seems much quicker (less than 20 seconds) than the protracted ritual of lethal injections