Should a connection be made with the Leonard Skinner mentioned in the Allan Sherman song "Hello Muddah, Hello Fadduh" and the gym teacher Leonard Skinner which the band Lynyrd Skynyrd took its name?

No. The song was written in 1963 by Chicago native Allan Sherman along with Lou Busch. The Leonard Skinner of Lynyrd Skynyrd fame was a gym teacher at Robert E. Lee High School in Jacksonville, Fla. who "tortured" his long-haired students, among them Ronnie Van Zant, Gary Rossington, and Bob Burns, who would go on to form the band.
From a 1977 Associated Press feature about Lynyrd Skynyrd:
"It seems a physical education teacher named Leonard Skinner didn't cotton to long hair or loud music. A run-in with him helped get the boys suspended. As a way of getting back, they named the band for Skinner, changing the vowels to avoid a lawsuit and becoming famous enough to make the story a rock legend."
https://en.wikipedia.org/wiki/Leonard_Skinner