I'm not really a fan of the character. But he could work in TDKR, just not as Batman's replacement. They could tweak it to make him just a badass vigilante on the scene that takes things much farther than Batman would and Batman has to bring him down because they have very different views, Azrael's being far more extreme. The difference is he actually poses a threat to Batman unlike the batman wannabees in TDK.
Batman doesn't need to be out of commission for that to work, nor does he have to take over the batman mantle (which is lame)
But that could very much work.