The stag button does not have a slot where you can place a piece of equipment - I tried shift + right click to see if anything opened both on action bar and in spell book. The only place to place mount equipment is the one slot in mount journal.
I don’t think they will apply it - it wouldn’t really make sense since we just auto-shift to swim form when we hit water. I think the same thing happens when someone is riding you, doesn’t it? Been a long time since I carried anyone in stag form.