Haguromo_ottsutsuki
Assume Peter Parker's origin is exactly the same.
Raised by his Uncle Ben and Aunt May. A social outcast/nerd at Gotham City High school, he gets bitten by a radioactive spider on a field trip. While trying to achieve fame, he lets a burglar go who ends up killing Uncle Ben, and Aunt May commits suicide. With his mantra "With great power comes great responsibility ", he becomes the Amazing Spider-Man, protector of Gotham! His personality is brave, selfless, and mysterious. Yeah just like our resident Billionaire .
Meanwhile, Bruce Wayne/Batman is a prime veteran at this point. The concept of Robins hasn't been made yet. Bruce catches a whiff of a new vigilante who operates day and night with powerful capabilities.
How will he react to Spider-Man? Would he allow him to do his job in peace or tell him to stay out of the way ? Would he want to train him?