In my experience, the reasons for planting this responsbility in one department or another are historic; i.e. unique to the institution. As Edie notes, it may even be the background of a particular employee that prompts their migration from one area to another.
If the proposed regulations requiring banks to identify protected federal payments are finalized, then federal functional regulatory agencies will be responsible for examining for compliance. (Yeah, like they know something about the topic.) From my perspective, that puts a pretty big weight on the "compliance department" side of the scales.
In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.