Oh! I left out the part where we see that Bruce made that costume for himself and wore it for a time when he was a kid.
So he's got it handy and is passing it down. Costumes are a big element in this story, because it's about Thomas Wayne's original bat costume, then the Robin costume.
Personally, I do prefer that it's a Flying Grayson's costume, myself. It's one of the two things I really am not crazy about in this story, the other --that I like less--being the revisiting of a change in the violence visited on Bruce's parents--a change from a random mugging to specific murder.
But in fairness to the writer of this story, both elements (the murder versus mugging and the 'Bruce as a kid wore the Robin outfit while he learned Detective work') were there in 1950s stories, and then got re-included in 'Untold Legends of the Batman'.