I honestly have no idea what claims Canada makes to the Arctic north of Alert. There is no landfall in that area. Just the icecap.
There are probably parts of the passage that Canada can claim soverignty over, however, there is still the principal known as "innocent passage" If necesscary to pass between two bodies of international waters, a foreign nation's shipping, including warships can transit as long as they are just transiting. They can't stop and do manuvers, exercises etc. It is like an international easment. If a foreign ship is doing more than "innocent passage" the nation with the territorial claim can order them to leave or take action if they don't. However, one of the fundamental principals of innocent passage is that you don't have to get permission as long as you comply with the law.
During the cold war, the Russians actually damaged a couple of our ships transiting the Bosphorus straights in a freedom of Navigation dispute. Our ship's still freely go in and out of those straights.