JP brought the black gorilla tape with him for a purpose. It might have been part of his plan in case he needed it, or part of his fantasy on how the crime would go down in his planning. One thing I think possible in trying to understand his thinking, he wanted to use that tape.
I can’t think of a plan he had in advance to have Denise put the tape around Jayme’s mouth, since he could not have predicted they would be hiding together in the bathtub.
Denise didn’t go as fast as JP wanted, so he put the gun down on the sink area, and did it himself, so another clue on what a rush job he had in mind he was ready to make a mistake. It still worked out for him, though.
Why he wanted mom to put the tape on her own daughter is suspicious to me of one of two things 1) JP is sadistic and got thrill from it or 2) JP didn’t want to put the shotgun down. He got frustrated and did anyway.