I'm sure they were Iranians. As you say, they had no need to get out the same way they got in, so obviously they weren't strangers to Karbala.
It was very carefully pulled off, as if they had some kind of definite objective other than just mayhem. If they had wanted to do that, they simply would have bombed the place once they got near enough.
Perhaps there was someone there, or they thought there was going to be someone there, whom they had intended to kidnap. Or maybe it was to get some kind of information they thought was being kept there.
Don't think they were Iranians. They were Iraqi Shia with big Iranian support. This operation cost alot o' Dinars. The Iranians don't do their own dirty work in Iraq. They buy it. They use Iraqi Shia proxies. Iranians, as evil as they are, are too damn smart not to create cutouts between themselves and the operators. Ahmed Chalabi is an Iraqi Shia and Iranian agent. The Iranians play a very smooth game when it comes to false flag operations.