Well the Rico model on the obelisk altar is pre-fight I believe. But if you're saying going into the boss area with a Red Ring equipped I'm not too sure that's a trigger (if there even is an exploitable one). Maybe it's just randomly generated when loading into the area / when defeating Olga. Or maybe it's linked to the current beat timer. Either way it's still awesome.
Edit. If the Rico model is only available when RETURNING to the boss room then maybe that could sustain a theory of defeating Falz while having a Red Ring equipped.