The Denmark Strait battles

Dunkerque and Strasbourg vs Bismarck and Eugen would be pretty dicey (pardon the pun) for the French I suspect. Especially if Bismarck could stay beyond the range of the French guns. Dunkerque and Strasbourg vs Scharnhorst and Gneisenau seems to me a pretty even match. Hoods spectacular explosion aside, a gun duel between battleships would likely be a prolonged affair. Look how many shells it took to reduce Bismarck to a hulk. One side or the other would undoubtedly retreat once the match got too one sided. Like PoW finally had to make smoke and pull out.

The big thing is what would the strategic outcome have been? In any of those matches both sides would likely take enough damage to require a return to base. The side that "loses" is the side that if forced to abandon their offensive mission. I suspect that is why Lutjens was too stubborn to return to Germany after he destroyed Hood even though Bismarck was damaged and losing fuel. If he had gone back to Germany it would have been a tactical loss for Britain, but a strategic win. In the end it was a strategic win for Britain anyway since Bismarck was hunted down and was never able to actually attack a convoy. It was a very costly win, but a win never-the-less. Lutjens should have taken his tactical victory and run home while he had the chance.

Quote weeds.

Yeah, the Battle of the North Cape is another one with prolonged punishment as the solution to the question of defeating a battleship with big guns.

I think Dunkerque and Strasbourg would have had a chance to disable Bismarck to a position that they could set the range they’d like and enlist help from allied ships to finish her off like the British did. If the premise is that France has its full navy at the time of the operation they could call on the rest of Force de Raid out of Brest and any available land based aircraft that could reach. The shortcoming I see in their line of battle is a dearth of aerial torpedoes.

I suspect some of that stubbornness was, "for the dispatches," so to speak.  Keep in mind that he had unreasonable bosses. 

That would be an interesting battle.  What escort did Bismarck and Prinz Eugen have at the time?