Barry Manilow checked out the Broadway competition while in town searching for a sound designer for his upcoming musical, “Harmony,” last week. But while the “Copacabana” singer caught “Kinky Boots” and “Cinderella,” one show he did not catch was former pal Bette Midler’s “I’ll Eat You Last.” Manilow and Midler were once collaborators, then reportedly fell out, so perfectly plucked eyebrows along the Great White Way were raised when Midler was missing among the stars in the audience at the recent “Manilow on Broadway.” A source told us Manilow did request tickets for Midler’s current one-woman show about agent Sue Mengers but got rejected when, “He was told they didn’t have any tickets for him.” Manilow’s rep tells us, “Not true. Barry never asked for tickets. He had limited time and went to a few shows because he was holding auditions for ‘Harmony.’ ”