We don't have synonyms for badmitten yet — it may be a rare, specialized, newly coined, or misspelled word. The closest thesaurus entries we have are admitted, besmitten and badminton.
(n) :
Misspelling of badminton. [(uncountable) A racquet sport played indoors on a court by two opposing players (singles) or two opposing pairs of players (doubles), in which a shuttlecock is volleyed over a net and the competitions are presided by an umpire in British English and a referee in American English.]
(n) :
Misspelling of badminton. [(uncountable) A racquet sport played indoors on a court by two opposing players (singles) or two opposing pairs of players (doubles), in which a shuttlecock is volleyed over a net and the competitions are presided by an umpire in British English and a referee in American English.]