Bismarck had some very sound reasons to deny, both geopolitical and internally ... UK would very quickly stop being Prussia's ally if they suddenly got all the coastland of Denmark (with its good shipyards), and the distant settlements in Faraoes, Iceland and Greenland, and the colonies in West Indies, not that UK would go to bed with France right away, but they would certainly try to restrict Prussian power, since it would much faster be obvious that Prussia is the strongest continental power (and UK have a 'tradition' of allying the second strongest, trying to keep the strongest down)
Internally, as touched upon, Denmark is too culturally distinct when compared to the rest of the Germanic territories since they speak a distinctly different (non-mutually intelligible) language,doesn't have a strong history of being part of HRE (just some 40'ish years just before the turn of the millenium), which, as dead as it was, still was the guideline for which areas were German, and a distinct concern that it would be a hotbed for rebels, both due to the distinct otherness (and the fact that Denmark was 'somewhat democratic' and the population weren't in on the deal), and due to the fact that it would be painfully easy for hostile nations to ship in support