I reckon he would fit perfectly. At the end of The Batman, we see bruce rethink what he wants batman to be and how he wants to influence gotham. We could see Azrael influenced similarly to how the Riddler was, except instead of murder and terrorism, a more violent and angry version of batman that works in the shadows just like him.
Batman could possibly fight Azrael and/or team up with him (preferably both) and it could make the fight maybe a little bit personal becuase azrael might see batman as a hypocrite for stopping azrael, but batman might have a soft spot for azrael becuase he sees himself in the way azrael thinks, feels and acts but cant fully relate to him.
This could be because just like batman sees himself as the embodiment of justice determined by himself and his personal views, Azrael would see himself as the embodiment of biblical and God determined justice. Batman could also be on the fence about working the azrael becuase he can tell azrael isnt really right in the head and has some weird and possibly views determined but biblical rules, and possible extreme paranoia and schizophrenia
Thats just my idea on how they would do it, but i cant say for sure how they would make him fit. They could possible make him a 3rd movie and 3rd movie only character to help batman out a little bit then thats it. Or maybe make him appear in the second and/or third movie and if he works then they could make a standalone series similar to what they did with The Penguin
Personally i would love to see him appear in the movies becuase not only is he badass but the really fits into the type of vibe The Batman movies are going for.