I had wanted to stay in my thread, but maybe I'll get more traction speaking in here.
My suggestion is a Head of State (a President?) elected by the Parliament with a 2/3 majority who serves a single consecutive term of 3 months. That way, you could have interesting dilemmas and the like if the Parliament failed to elect the President on time (such as an automatic dissolution procedure?)
As a fair warning, I'm unaware of any country that demands a supermajority in parliament to elect a president and has not entered a political crisis over it.