Transgender Reporter Zoey Tur -- Accused of Battery ... Over Caitlyn Jenner

Going to battle over Caitlyn Jenner could get transgender reporter Zoey Tur arrested -- a guest on Dr. Drew's show has filed a police report against Tur for getting handsy during a debate.

The heated convo went down Thursday on “Dr. Drew” ... Ben Shapiro is accusing Tur of threatening him on the show.

If you missed it ... Tur said if Shaprio didn't get his pronouns in order he'd be taken home in an ambulance, while putting her hand around the back of Shapiro's neck.

Shapiro went to LAPD on Sunday to file a criminal complaint.