1- Why does this require a transition to a federal empire?
2- What could make the British persue this goal? It was incompatible with their tradition balance of power policies.
1) Up til then the existing meme of empire was 'center rules everyone'. If the Brits had managed to put together a 'we've got a system where the peripheral entities are a constructive part of the empire and are happy with it', then the Germans are more likely to want to follow that example. It would be 'safer'. OTL, many smaller German states recognized that Prussia would be overwhelmingly dominant, and were very reluctant to join.
If Britain remains the unitary empire she was and remained OTL, she would not serve as an inspiration/a nucleus for a German empire
2) Again, with a 'federal'ish empire, Hanover is likely to be more caught up. OTL we had big center, several minor peripheral bits of little consequence (the various colonies, Ireland, to some extent Hanover). Hanover and Britain might feel more fraternal, less like step-children under one 'father'. Hanover, with the friendly support of Britain, would be a less threatening nucleus for the formation of a German nation than either Prussia or Austria (the two OTL candidates), while being large enough and strong enough to be a possible nucleus.
Balance on the continent is fine if you can't have direct influence. Being the Emperor of the HRE and ruler of one of the major sub components would give them that influence, and could easily have them change their minds.