Instead of directly killing Lennon, Chapman stalks him. After a few incidents, Lennon and Ono actually start to fear for their lives. They're uncomfortable with guns and guards, but the police don't appear to be doing a very good job trying to find this guy.
A good friend drops by to loan them a pistol, for their own protection. They tell him they don't want it, but the friend forgets it (intentionally or unintentionally) on the table when he leaves.
A couple of hours later, Lennon sees Chapman out on a ledge outside the window, having climbed up the building. Frightened, and fumbling for something to protect himself, he accidentally causes the gun to go off, shooting Chapman.
Another option: Chapman kills Ono instead, but gets off on a technicality. Lennon, grief-stricken and angry enough to forget his pacifism, talks briefly to the mob, but in the end decides not to do anything anyway. However, a communication failure leads the mobster to misunderstand some Mancunian idiom as a statement to go ahead and finish the job.
Sorry, all I've got are accidents, I guess.
EDIT:
One more idea:
John Lennon, a taxidermist from Oklahoma with a history of mental illness, is getting tired of this other guy stealing his name. He goes to New York to kill him, and arrives just in time to see Chapman holding a gun on musician Lennon. Not wanting someone else to do him in first, taxidermist John Lennon shoots Mark David Chapman.