The Murmansk convoy run has the advantages (for Germany) of being far from Allied bases, relatively close to German bases and offering good targets. The disadvantages include the weather, which is awful much of the time and which makes aircraft operations difficult-to-impossible even for experienced carrier crews. Also, the Royal Navy is seriously over-stretched but prioritizes defending those convoys and making ready to take down Tirpitz.
The two serious battles (Barents Sea and North Cape) were fought at night and/or in 'limited visibility' (IE in a blowing fog-and-snowstorm with visibility limited to a little above 'what did we just run into'). The actual and planned sorties of Tirpitz offered some potential use for a carrier (if it had planes, and they had weapons)... except that those convoys were escorted by one or two battleships and at least one carrier.
I don't give Graf Zeppelin much of a chance against one or two Royal Navy carriers. If they get just one torpedo hit, GZ is likely to not make it home - and if she does, she's never getting back to Germany for repairs.
Well, I meant, what if the GZ never went to the high seas, to the Atlantic, neither to the Arctic Ocean, but instead, would have been stationed at the Baltic Sea and the sea area between Sweden and Finland, the Gulf of Bothnia?
Baltic Sea campaigns during 1939-1945 were very limited. The German Kriegsmarine and the Finnish Navy had the upper hand. The Soviet Baltic Fleet was isolated and not actually made any significant operations in this area, until Finland's exit from the war, in September 1944.
This sea area was never a main theater in World War II. However, if the GZ had been brought to the Baltic Sea, the entire Arctic theatre could have experienced major changes. The British Royal Navy avoided the campaigns at the Baltic Sea, there actually wasn't a lot to do there, except the one operation, ultimately abandoned by mr. Churchill.
Operation Catherine was a proposed Baltic Sea offensive by United Kingdom's Royal Navy in the early WWII. It aimed to end Germany's commerence with the other Baltic Sea nations, especially, to stop the flow of Swedish iron ore to Germany and prevent Sweden from joining to the Axis powers. However, the biggest problem for the British was, how the proposed fleet would have sailed to the Baltic Sea? Churchill was aware of the many risks confronting Operation Catherine. He knew German warships, submarines, minefields, and bombers would threaten the fleet. These problems might have been solved in the end, but at that stage of the war, they did not seem to be the most important of all, but it was felt that the priorities of the war were elsewhere. Catherine was abandoned.
The Kriegsmarine was the dominant naval power in the Baltic Sea between 1939 and 1944. If the GZ would had been stationed here, there was no enemy fleet, no other threats for that carrier nearby. At least, it would have required time, from Germany's enemies, to respond for this maneuver and meanwhile, the GZ could run "wild and free" at the Baltic Sea.
Since this sea area is also in the north, the conditions are challenging for flying, sea operations, especially 80 years ago. However, the weather in the Baltic Sea is more mild than in the Arctic Ocean. In order to attack, from the Gulf of Bothnia, against the Allied shipping convoys sailing in the Arctic Ocean, the Norwegian Sea, or the Barents Sea, the German Luftwaffe would have had to fly partly through Norway, or Finland. It would have been possible.
The German Junkers Ju 87R-2 was a long range dive-bomber, intended for anti-shipping missions. This model would have been capable to operate from a carrier. It had an operational range of 1535 km (954 mi). Attacking the Allied convoys in the Arctic would had been within its range. In addition, the Stukas could have made a stopover at Finnish airfields for refueling, maintenance, resupply of ammunition, etc.