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That's the weird thing, though; I had a gamepad connected and P1 was assigned to the keyboard and there was no evidence of P2 existing nor being able to control it. I guess I'll try a disconnect and reconnect later.

Did you press the select button on the unassigned controller and press right on it to assign it to P2? As long as the gamepad is an xinput device (0-3 gamepad ID range) the game will sniff for inputs to assign it from either the overworld menu or from the little widget that pops up if you press select on the controller