Another character who could be a great alternative to Batman is Owlman.
Just putting Jake Gyllenhaal side by side with Robert Pattinson is enough evidence I need to know he would be the right choice.
Batman and Owlman once crossed paths in the 2010 DC animated movie Justice League: Crisis on Two Earths, which saw the Dark Knight, this is not such a far-fetched idea.
Because (depending on which version of the character you are referring to) he is essentially an alternate version of the Dark Knight from another reality in the DC Multiverse, but with a more sinister agenda (keep note of that fact that owls are known to prey on bats).