The Monarch of Canada is a ceremonial position. The Monarch's representative (the Governor General) is appointed by the Prime Minister and has no real power (see: the King-Bing Affair for legal precedent), and therefore could technically be done by anyone from anywhere (even beyond the grave). Politically speaking, absolutely nothing would have to change. The Monarch's effective power in our political system would go from basically zero to literally zero, thus eliminating an avenue for potential abuse of power that we risk by keeping a living Monarch as head-of-state. We could achieve this without having to re-open difficult constitutional questions. Traditionalist Canadian institutions with "Royal" in their names (Mounted Police, Army, Airforce, etc) would not have to change their names or branding. Heck, we wouldn't even have to change the designs on our money. Literally nothing would change except closing a loop-hole (albeit a very low-risk one) for potential power abuse in our political system.
The only down side is that smug know-it-alls can say "actually Canada is not a democracy, it's a Constitutional Necrocracy"
Since the Governor General must have real power, it follows that the Monarch must have real power regarding the appointment of the Governor General - rejecting the Prime Minister's request to dismiss or appoint a Governor General when this is clearly an attempt to fill the position with someone who will allow the Prime Minister to act non-democratically.
A dead person will not be able to fulfill this role.
Counter-hypothetical: what if the Monarch decided to act against the Prime Minister and appoint a Governor General to act against their mandate? Both your hypothetical and mine are incidents of "bad-behaviour" going against norms to push agendas. We would prefer were that neither were possible. However in your hypothetical at least the person exhibiting "bad-behaviour" (the Prime Minister) has some mandate given that they were democratically elected. Whereas in my hypothetical the person exhibiting "bad-behaviour" is an inherited position held by someone in lives in a far-away place and may have only set foot in the nation they are meddling in a handful of times.
In either situation we're accepting the risk of bad-faith actors manipulating the structures of power, but if we ditch the Monarch, at least the person doing so is in someway accountable to the people. Harper was successfully able to dodge a confidence vote, but in the end he was ousted from power in a democratic process. I'd argue that's the better scenario.
Could we not solve the problem of the PM appointing a lackey as Governor General with other form of check-and-balance that requires zero input from individuals with no connections to a democratic process? Perhaps a similar way that Supreme Court Justices are appointed (candidates recommended by the Prime Minister and approved by the federal cabinet). While not immune to abuses of power, I would like this better than a Monarch being that check-and-balance.
In the unlikely event that the UK were to abolish or deprecate its monarchy, Canada would still prefer not to re-open the Constitution. This might indeed lead to Canada worshipping "The Crown" without anybody to wear said ceremonial headgear.
In other words, the logical contortions of a democracy naming one family as being more important than anyone else, and it being a family without power anyway, are less painfully absurd in Canadian politics than discussing the Constitution. ^_^