It’s starting to look like filmmaker Dori Berinstein is carving out a niche for herself with docus that appeal to theater fans, as new doc “Carol Channing: Larger Than Life” follows on the heels of “Showbusiness,” a chronicle of the run-up to the 2004 Tony Awards.The overlap, though, is a natural one, given the fact that Berinstein’s been producing on Broadway since the 1993 outing “Full Moon.” In fact, it was as a producer of the stage version of “Thoroughly Modern Millie” — the 1967 film in which Channing co-starred — that Berinstein got to know Channing. “I didn’t seek Carol out to make a film,” she says. “It was really through becoming friends with her that it became clear there was a great story to tell.” The docu chronicles Channing’s work over her long career, with talking-head contributions from Jerry Herman, Chita Rivera and Lily Tomlin, among others. One sequence, filmed last year, sees Channing run into starstruck cast members of Rialto musical “Memphis,” mid-matinee, in Shubert Alley. The film’s most inspiring and touching moments follow the unlikely tale of Channing’s rekindled romance with her childhood love, Harry Kullijian, whom she married in 2003 after decades of separation. Reaction from auds at the Tribeca fest, where the film premiered, has been positive, with the sales agent said to be already fielding offers to acquire the distribution rights. Pic bowed April 23 to a receptive house that included a number of Main Stem dancers who had worked with the leading lady. The actress herself, though, couldn’t make it due to a minor illness that kept her in California. Instead Tyne Daly introduced the film by reading a note from Channing — but not in that signature voice. “I don’t do Channing,” she warned.