Why did the British begin to feel that pursuing the war further would not be beneficial to the Empire overall?

Assuming that you are talking about the end of either WW 1 or WW 2: first of all, the war simply had been won already. Other good reasons for not pursuing the war any further: 1) the wars had cost Britain billions of dollars an by the end of the war Britain was practically bankrupt, and 2) the Empire nations had seen their overlords being clobbered on the battlefield and having become dependent on the Colonies' soldiers. Which was already leading to more and more British colonies clamoring for self-rule or independence.