Perhaps, but they might have kept the British destroyers busy, and possibly some of the cuisers, too, leaving SCHARNHORST free to concentrate on DUKE OF YORK. I agree that the outcome would likely have been the same, but the Royal Navy would have had to work a lot harder for the victory, IMHO.
i don't thin that the Britisch would make a great deal about working harder, once the scharnhorst was sunk, the convoys towards russia would be safe (at least from enemy surface units). by the way, why was the Gneisenau kept so long in dock and later just left over there? what if the scharnhorst and gneisenau were together fighting at the battle of the nord cape. any chanche the DoY would be sunk?
Gneisenau was to have rebuilt with 15" guns in twin turrets but after removing bow, old turrets and various other components, work was abandoned. And against DoY: When S&G met Renown 1940, they ran. Seriously: Althought S&G were seriously undergunned, they had theoretical 8 barrel and one hull advantage against DoY. I think this comes down to the question: will that radar-crippling hit occur to Scharnhorst as historically? If yes, I think S&G are in serious trouble. If not, they are in trouble.
Both SCHARNHORST and GNEISENAU hit mines during the "Channel Dash" in 1942, when they left Brest to return to Germany and caught the Brits totally flatfooted. GNEISENAU underwent repairs, but the RAF kept bombing her in the stocks and the Germans finally gave up on her, eventually scuttling her hulk as a blockship.