Landis seems to me to have conflated things he did pick up with something he read in "Six Seconds in Dallas", a 1967 book given him in 2014.
"By this time someone was lifting the President's body out of the right side
of the car. Agent Hill helped Mrs. Kennedy out of the car, and I followed.
Mrs. Kennedy's purse and hat and a cigarette lighter were on the back seat.
I picked these three items up as I walked through the car and followed Mrs.
Kennedy into the hospital."
-- Landris' statement, Nov. 27, 1963
In the 1967 book, Josiah Thompson advanced the theory that the President's back wound was shallow (due to a weak round) and that the bullet might have worked itself out. Thompson implied the bullet worked itself out onto the President's stretcher at Parkland, so that's where Landis left his "bullet".
Thompson later backed away from the shallow wound claim. Even in his book, he writes:
"It might be argued that the character of this wound is not fully known.
It was explored with fingers and metal probes, but it was not dissected.
Here is an existential problem that no number of autopsy photos and
X-rays can clear up."
Thompson also didn't believe the jacket and shirt material had a bunch at the nape and so dismissed the neck transit.
The "shallow wound" and "non-transit" conspiracy myths, while powerful arguments at first glance, might have influenced Landis.