The key fact that stands out to me about Barrett's claim--which is where it started--is that he never mentioned it until the mid-1990s. No one did. No one mentioned or breathed a word of having seen an Oswald wallet at the scene for over thirty years! That does not pass the smell test of it being Oswald's wallet, which would have been extremely memorable and the wallet logged into evidence with the Crime Scene personnel (Doughty and Barnes) right there at the scene. But it wasn't, and nobody ever heard of that being Oswald's wallet for another thirty years, because it wasn't. Croy's signature saying he found Oswald's wallet turned up posthumously and he never mentioned anything of the kind in his lifetime. If he did come to believe that at the time he wrote that photo inscription--and was not writing it as a joke--he may have become convinced it was but nothing in his reported memories of that wallet confirmed or supported that it was Oswald's.
The wallet is in the hand of an officer holding Tippit's revolver, and there is a report of Callaway being divested of Tippit's revolver at gunpoint in a confrontation, following which that revolver and Callaway and Scoggins returned to the crime scene. It is very reasonable that the two men who took the Tippit revolver from Callaway demanded his wallet, or Scoggins', one of those two, as "security" before allowing them to drive separately back to the crime scene. The wallet was then given to police at the scene with the Tippit revolver, police checked it, realized Callaway and Scoggins were innocent, and the wallet was returned as of no further relevance or interest in the crime case, which is why it was not logged in as evidence. If Callaway didn't volunteer that detail in later years it would be because he is known to have been sore and sensitive about the embarrassment of that confrontation, which would be the background on that. The similarity in appearance to Oswald's wallet would be attributed to coincidence as being a common type of man's wallet sold.
Barrett thirty years later may have confused his memory in when and where Westbrook asked him that day about the names of Oswald and Hidell. Myers makes a good case for that as the explanation. But another possibility is the case against Oswald on Tippit was actually weaker than it has been popularly supposed, and for all we know Barrett could have planted that Oswald wallet story as intent to shore it up by introducing a new smoking-gun incrimination of Oswald that would remove any doubt (if there was any). (He told fellow FBI agent Hosty who believed Barrett and published it in Hosty's book, is how Barrett launched it.)
Barrett is the same FBI agent who was a witness of Oswald's arrest and later told Myers with a straight face that the way Oswald got his bruised black eye was Oswald had swung his face into the hand of an officer, giving himself a black eye by assaulting the officer's hand with his eye, lol. That's an aside, sorry. The relevant point is one cannot take seriously incriminating evidence newly introduced from memory for the first time 30 years later, that nobody at the crime scene at the time ever heard of (referring to Oswald ID). This is how urban legends start, which is what this one is.
If there were earlier or contemporary credible witness reports of Oswald ID in that wallet, that would be a different matter. But this story starts for the first time 30 years later. And the wallet wasn't handed in as evidence at the crime scene on the day of the crime, and it is seen in the WFAA-TV film in the same officer's hand as the Tippit revolver which associates it with the hostile at-gunpoint confrontation with Callaway in the obtaining of that gun (and probably Callaways's wallet too).