The same reason he attacked France in the first place, power. The Danes have a significant level of trade under lock and key through the Baltic. Furthermore, the main reason for Henry was determined to reclaim the lands held by England prior to the Hundred years war, this was the cause of many of his attacks on France, and Scotland too, especially at Flodden field. At no point was England truly ready to fight the French in this way, even with significant aid from others on the continent.
Henry VIII was very much a fool in this regard.
I understand that the reformation took place in Denmark after the English had already turned protestant, but England was very much a forced event. Henry enforced it and any who opposed died. In Denmark I believe this was less true, with a much more natural process leading to reformation. I may be wrong about this, but Denmark was starting to stray from the Catholic church at this time.
My main question is not about the why, quite frankly, if Henry VIII wanted to invade he was willing to make up a reason for it, and as he was a favourite of the pope before his departure from the church, he could get away with it quite often. Pretext for war could range from trading rights and privileges to religious differences, especially as Fredrick 1 claimed religious freedom from Rome in 1523. The question I want answered is whether it would change anything.
Thank you very much for the help you're providing.