I think it's the notion of getting the kid into the Blade Runner organization unnoticed that's a bit too difficult to believe for him. Which frankly, does make sense. Nexus 9 models aren't made as children, are they? At least, if we're to take that one scene with Wallace and that replicant-from-a-bag as the norm. So really, if they are just ready-made adult human beings at "birth", and are immediately shipped off to the LAPD, it'd be pretty much impossible for him to have ever lived in that orphanage, or even to have ever been a child. Unless of course, K's presuming that there are people working on the inside of the LAPD keeping him safe or whatever. And that could be a potential explanation as well.
But again, I can easily explain this away with K just really wanting to believe that this was true, as well as being convinced by the evidence that he had already collected. I think the difference between that guy and me is that I'm just going along with the flow and thought process of the main character, while he didn't.