Based more on the original 1930s comic book version of Batman than on the later films.
Salazar Gaunt knew that he was a squib. He was not even a parselmouth. He was a clever little boy, unlike his more magically talented, but moronic brother, Morfinn. So he ran away, changing his name to 'Bruce', which he had seen on a gravestone in Little Hangleton graveyard. It being the time of the Great Depression, he managed to mingle with the muggle children being shipped off to Canada, and because of his devastatingly good looks was adopted by a couple of childless muggles, as well off as the muggles who lived near the Gaunt shack, and became known as Bruce Wayne. They moved too Gotham City in America.
The Waynes were very wealthy indeed, but it did not stop them dying in the violence which swept Gotham City a few years later, leaving Bruce a very wealthy, but angry and bereaved little boy, having seen them murdered. He had no magic with which to retaliate, so he concentrated on making himself as physically perfect as he could be, whilst remaining up to date with the magical world, which was easier under Macusa than in England, fighting crime as the Dark Knight.
A prophecy that one of a pair of twins would tear Britain's magical world apart in jealousy and anger led him to seek out his sister's children in an orphanage. One of the two boys she had given birth to was a squib, and knowing that Tom Marvolo Riddle would get his invitation to Hogwarts in due course, Bruce adopted the squib sibling, Robert Morfinn Riddle, calling him 'Robin' and raising him to be his helpmate, assuming that it would be a squib who would feel jealousy and anger against his more talented brother.
A tragic mistake.
Not sure where this is going, but it struck me as having possibilities.