About the Live Team Match API (https://api.chess.com/pub/match/live/{ID}) I think there is an error when defining the time_class tag in rapid live matches. Assuming that rapid games have a time control longer than 10 minutes, the time_class tag is currently 'standard' instead of 'rapid'. It can be quickly checked from the very example for the API:
{"@id":"https://api.chess.com/pub/match/live/5861","name":"Friendly 10|2 Rapid Open: Srbija Tim vs Team USA Live","url":"https://www.chess.com/club/matches/live/5861","start_time":1579988425,"status":"scheduled","boards":0,"settings":{"rules":"chess","time_class":"standard","time_control":600,"time_increment":2,"min_team_players":1,"min_required_games":0,"autostart":false}
However, this tag is correct for lightning and blitz live matches.