It is worth remembering that the Bismarck-class battleship wasn’t intended to go toe-to-toe with the Royal Navy as the Kriegsmarine was well aware that they didn’t have the numbers to defeat the Royal Navy in a pitched battle. Bismarck and her sister ship, Tirpitz, were going to be used to raid convoys in the Atlantic. This created an issue for the Royal Navy for several reasons.

First of all, convoys in the Atlantic were being ravaged by unrestricted submarine warfare and destroyers were required to act as escorts as they possessed the speed and weapons required to effectively hunt U-boats. But a destroyer is no match for a battleship, so if the Bismarck could make it into the Atlantic then there would be the possibility that the Bismarck would coordinate with Wolfpack’s to destroy convoy groups and their escorts.

Secondly, if the Bismarck did make it to the open ocean and far away from any land installations, then air support would not be readily available. The reason battleships are no longer in use today is because air power demonstrated how effective a squadron of aircraft are at destroying battleships. Being the better option when it comes to cost effectiveness, spend years and a gargantuan amount of resources building a battleship or invest a fraction into a squadron of bombers. But in the Atlantic the only air support that could be deployed would have to come from carrier groups, which were expensive and several would be needed to cover the entire Atlantic in the search for a single battleship.

Thirdly, the fact that the Bismarck sunk the HMS Hood, the flagship of the Royal Navy, had two major impacts on British planning. The admiralty was now very concerned that the Bismarck was out there and demonstrated its ship killing capabilities, particularly at a time when Britain needed every available ship for the defence of the vast imperial shipping lanes. And the desire to avenge the loss of the Hood would definitely have influenced the decision makers and their opinions when it came to plans to deal with the Bismarck.

I am not a naval historian so I wish I could give you more details, or perhaps someone else will comment and provide more insight into areas which I do not know. But to answer your question, why did the British expend so many resources in the hunt for the Bismarck? Because it would have been cheaper in the long run to destroy the Bismarck as quickly as possible to prevent it from weakening convoy supply routes to a state which was under siege from both air and sea. In my own conclusions on the subject I believe that the cost effectiveness of focusing on taking out the Bismarck were the primary drivers behind the British decision makers but others may reach a different, yet equally valid, conclusion.

Sources:

Campbell, John (1985). Naval Weapons of World War II

Garzke, William H. & Dulin, Robert O. (1994). "Bismarck's Final Battle". Warship International. International Naval Research Organization.