A secondary British objective was to sink Scharnhorst so that she couldn't threaten future convoys - but keeping Scharnhorst at arm's length without sinking her would still have been a victory for the British.
This is not correct. The British convoys to the USSR were in range of land-based aircraft. The relative ineffectiveness of aircraft against them was the primary reason these convoys were even possible. At least during Winter: the British stopped running Summer convoys in 1942 and 1943 after the PQ17 disaster. A German carrier would have been even more constrained in its operation than German land-based aircraft.
First, you are overestimating how many good days were actually available, given the weather conditions in the Arctic for a a convoy that would be at sea only for about a week and of that out of range of UK and USSR land based air only two or three days.
Second, it matters a lot if those five days are not five days of stationary invulnerability but five days of running from threatening forces in a confined area. In five days a ship making 30 knots will cover 4,000 miles. That puts your carriers somewhere in the middle of Africa. Realistically running means returning to port and not being able to leave again except under observation.
So after PQ17 the next convoys defense was organised around a bunch of destroyers and a CVE. All heading into the artic in September where the CVE would be useless and should just roll over and die because of German big guns waiting in Norway.
Except that the fighter cover provided was super useful and the biggest problem they encountered was a lack of defensive fighters. No big guns were used on either side.
Convoys need destroyers and CV/CVE fighter cover to fend off print subs and AC.
Fleet engagements are still settled between the big boys when they come together for whatever reason.
The last place BBs were really useful was in forcing or breaking up a landing action when there was true parity on naval power. If you can force the carrier's into a defensive posture then your SAG can get in among the invasion fleet necessitating a brawl.
This being said, I don't know a lot of people who would attempt a naval invasion without control of the sea lanes...