I'd say abolish the monarchy outright and have a full Republic like they had in the 1600s after they beheaded Charles I, minus the whole "Lord Protector" Cromwell type figure. Most European countries have successfully switched from kingdom to republic. The exception is France, we could reverse the situation there and crown the Duke of Anjou as King Louis XX ( https://en.wikipedia.org/wiki/Louis_Alphonse,_Duke_of_Anjou ) it would be worth it if it meant getting rid of the awful French Presidents forever.
There needs to be a head of state. The Regent would be no different that the Presidents of Ireland, Germany, or Italy.
My plan would call for only a modest budget and small honorarium.
Avoids having to change the structure of the government, a divisive debate on republicanism (that unlike Australia, almost no one on the right over there supports, so it would never be on the table unless the government was hard-left), and millions in administrative costs in changing the name of the country.
Franco technically restored the Spanish Monarchy and made himself regent for life.
Crowmell was King (and was even followed by his son! Though he refused the crown ala Caesar) but in name so I’d dispute they were ever “really” a republic.
Re Grania’s point, I don’t see why, given the nature of the position, there would be a problem getting someone approved. Canada and OZ don’t see to have any trouble picking Governors-General. If need be ditch my supermaj requirement.
The alternative would be what, have the PM directly elected by the Commons and having them be Head of State ala the President of South Africa?