Well for starters Harry would be a blood relative in this scenario as well as an infant at the time, so he might take a different approach out of respect for his sister and how she might want him to be raised. That likely wouldn't stop him from training him before his abilities manifested fully - Bruce would be well aware of the dangers of the world into which Harry would ultimately become a part of so I could see him training him to be a critical thinker and capable of self defence and hand-to-hand combat from an early age though I see him softening somewhat in his approach. Harry would have an excellent relationship with Alfred who would effectively be his second father and assuming that Dick came into the picture later on and was around the same age as Harry I could see those two bonding through their shared trauma and if they discover Bruce's identity as the Batman may even want to join him by their own choice as his dual sidekicks, likely having to pressure Bruce to go along with it. With this version of Batman having a direct personal connection to the wizarding world I could see him becoming more educated in matters of magic than the OTL version, likely maintaining contact with Dumbledore behind the scenes and being one of the few Muggles that he fully trusted. While not possessing any magical ability himself I could see this particular Batman having an encyclopaedic knowledge of the magical world and being particularly adept in the use of certain magical items, prepared potions and artefacts to such a degree that it puts him on par with most wizards he might come up against. Any Soul Eater coming up against him would drastically underestimate him as a mere Muggle only to quickly fall to his superior knowledge and preps against magical attack. Voldemort would present a challenge once he took the Batman seriously as an enemy however I could ultimately see the combined efforts of both Bruce and Harry working together being what finally brings him down.

With Voldemort defeated and Bruce's skill and ability widely recognised by the wider wizarding community I could see them giving special dispensation to him as a friend of their world to not be obliviated (like they could anyhow) granting him equal standing in their world as both a father to Harry and as a capable Muggle who can be relied upon in any situation.