I'm not thinking a rotating monarch (though there are things in that ideal that appeal to me) but a system where we have Co-Heads of State. The Regeu and the Schac'h share powers, but are chosen by differing methods. The Regeu would be the current King, while the Schac'h is chosen by election periodically (I was thinking every 2 years? Or maybe whenever a complete cycle of Senats elections happens).
I see legislative powers alternating (Regeu gets odd Clarks, Schac'h gets even, as an example), with a veto from the "inactive" member possible. This would sate a certain Republican impulse while retaining the King. Plus, it's Even More Novel than Monarchy!