If you're interested in the geeky side of this stuff, you'll find BP and Batman often paired in a 'who would win' DC vs. Marvel on sites where people discuss that kind of thing because of how they pair off in terms of wealth, tactics, tech, fighting ability, non-use of guns etc. and mindset.
You're referring to the fame of the characters, and there's certainly no contest there, but you'll not hear Ironman referred to as Marvel's Batman as often as the likes of BP, NightHawk or Moon Knight.
I've not seen a single trailer for the film, so I'll be going into it blind. There's so much potential and I hope it's a cracker.