There's something not right here. Do y'all really think it's credible that a Secret Service agent would put such an immensely valuable piece of evidence on a hospital stretcher? - where it could have easily gone unnoticed momentarily and dropped off, rolled under a cupboard, been fingered, stolen as a souvenir?
Following professional protocol, he would immediately have pocketed the bullet, then submitted it to be bagged as crucial evidence.
So, the discovery of the bullet on the stretcher points more to an "amateur" plant.