[Brin]

I believe that PatBC has the most critical point, the RN needed to stop the Bismark from breaking out and running free. She had excellent speed capabilities in comparison to most of the RN large ships of the time. I suggest that it was this aspect of speed that Lutjens wanted to use to break free and complete the task assigned to her.

As to reasons why Bismark didn't initially open fire, is more likely that it would have been completely obvious at the start of the encounter to all the participants that the Hood and Prince of Wales (PoW) were hopelessly out gunned and outclassed. It would have seemed absolutely pointless (bordering on suicide) to engage such a superior force directly in a ‘set piece battle‘, from the Bismark’s viewpoint it would be an engagement that would only slow down and therefore reduce any chance of achieving its aim. From the aspect of the Hood and PoW it was to follow/track and if possible harry the German ships. Although a mistaken target by the PoW over the Eugen for Bismark is possible, it is also unlikely. I would suggest that by skirmishing the Prinz Eugen, it might slow the group, thereby allowing sufficient time for the remainder of the RN forces to consolidate an attack. However if you start poking a sleeping tiger, and then when it wakes you find that you are to close…… is more likely why the RN ships then engaged the Bismark after the Prinz Eugen, but having to make the dash for range being much smaller gunned.
The Hood was a new (at the time) compromised design, reduced armour for speed. The gamble was exposed by the Bismark, although it could be argued only by a lucky shot.

As to why the Bismark did not sink the PoW, I would suggest that time was of the essence. For naval combat of WW2 usually involved timescales of hours and not minutes, for to the sink the PoW might take most of a day, as they were dealing with conventional artillery pieces mounted on a moving platform trying to hit a moving target of changing speed and direction which would be trying not to be hit. At which point Lutjens was most probably not aware of what contact the PoW had with other RN forces, and most importantly where these forces were located. So by leaving it to escape was strategically the best option in order to complete his mission.

The sinking of the Hood served as propaganda in Britain to polarise the nation's attitude, not to mention the resolve of the sailors at the frontline to find and sink her.

Visual recognition, most ships at sea look similar to those unfamiliar with naval aspects, but to the visual spotters and upper deck gun crews ship recognition was in effect their basic seamanship trade (as it still is today). It would only be by explicit camouflaging or altering a ship to appear as another could such a deception work. After all, the Bismark at 50,000 tons displacement, even if with a similar silhouette, by sheer volume appears much larger than the Prinz Eugen at 18,400 tons (max).

I felt sorry for the 100+ dockyard fitters onboard the PoW who were still refitting her during the sea trials she was undertaking at the time.

The point of vengeance, probably only appeared in the latter stages of the final battle prior to the Bismark being scuttled.