King Charles III is the head of state of Canada (King of Canada):
www.canada.ca
The Governor General exercises most of the Sovereign's power in Canada:
The role of the Governor General in representing Her Majesty The Queen.
www.canada.ca
The PM is the head of government...
As for the cost of the Crown, we don't pay tribute to the King but we do pay the GG's salary, residences, expenses, expenses of the King or designates while on Canadian business, the cost of hosting heads of state (dinners, etc.) and likely more under the cost of the "Crown". It is cheaper than paying to elect a President (to take the constitutional role) and then still paying ALL those expenses.