What in the world makes you think for even one second that the Hidell ID card was on Oswald's person? Paul Bentley was the cop who took Oswald's wallet from him in the car. The next day, during in interview on TV, Bentley was asked what was in the wallet and he answered something along the lines of "the usual things". He in fact mentioned a driver's license and a credit card. What he most certainly didn't mention was an Hidell ID card. In fact, none of the officers who were in the car mention a Hidell ID in their reports.
So they didn't read all the things in the wallet.
The first time the Hidell ID pops up is when Gus Rose (who just started work) talked to Oswald at the station. There, an unidentified person, gave him a wallet and said it belonged to Oswald. It was in that wallet the Hidell ID was found. There is not a shred of evidence this was the same wallet that Bentley took from Oswald.
You seem to be overlooking the testimony of Rose (7H228):
Mr. ROSE. Well, the first thing I asked him was what his name was and he told me it was Hidell.
Mr. BALL. Did he tell you it was Hidell?
Mr. ROSE. Yes; he did.
Mr. BALL. He didn’t tell you it was Oswald?
Mr. ROSE. No; he didn’t, not right then-he did later. In a minute-1 found two cards-I found a card that said “A. Hidell.” And I found another card that said “Lee Oswald’ on it, and I asked him which of the two was his correct name. He wouldn’t tell me at the time, he just said, “You find nut.” And then in just a few minutes Captain Fritz came in and he told me to get two men and go to Irving and search his house.
Mr. BALL. Now, when he first Came in there you said that he said his name was “Hidell”?
Mr. ROSE. Yes.
Mr. BALL. Was that before you saw the two cards?
Mr. ROSE. Yes; it was.
Mr. BALL. Before you saw the cards?
Mr. ROSE. Yes; it was."
And as far as the murder weapon is concerned; there is no evidence that the revolver now in evidence is in fact that one that was taken from Oswald at the Texas Theater. Detective Carroll said he pulled a revolver out of somebody's hand, but he didn't know whose hand it was. He then give the revolver to Detective Hill who subsequently walked around with it for at least two hours, showing it to the media and even giving it to Bentley (if I remember correctly) before he had a bunch of officers in the personnel room initial the revolver after telling them it was Oswald's revolver.
Well, Caroll said he took it from someone's hand in the Texas Theatre and kept it in his possession and put his initials on it before turning it over to Hill. It was not a police weapon and all but one of the people in the melee in the Texas Theatre were police officers.
There is also evidence that the revolver is indistinguishable from the revolver shown in the back-yard photos. There is also evidence that this was the gun that killed Officer Tippit.