WI: Frederik II of Denmark Marries His Mistress?

It seems this was a major fear during his youth that Crown Prince/King Frederik (II) of Denmark would marry his mistress, Anna Coritzdatter Hardenberg. Even once he became king, he pursued the course unsuccessfully, but eventually caved to pressure to marry a princess, namely Sophie of Mecklenburg-Güstrow.

What were the major reasons behind it? Wikipedia just says that Anna was already betrothed and that many of the king's ministers advised against a domestic marriage. But it also seems (from her wikipedia article) that Anna was ambivalent about the match, since she seems to have been rather relieved when the king left her alone, and she was one of the voices who encouraged him to marry abroad instead.

So, say Freddie decides to emulate Henry VIII and his cousin, Erik XIV over the Sound, and marries Anna Hardenberg. When is the most likely time this marriage would occur? Would there be children (Anna didn't have any OTL, but she might have been close to menopause by the time she wed OTL)? Would Frederik stay "king"? Or would the Danish nobility do the same to him that the Swedes did to Erik?
 
There wasn't a tradition in Denmark for Danish kings to marry common nobles, but in case he do marry her, the nobility will panic at first , but then they will decide, that it will make little difference. If Frederick and Anna produce children, it will mean a clearer succession than OTL regency council, which will be good news, if not Frederick have siblings/nephews to inherited him. It may create a tradition of Danish kings Harry Danish nobility once in a while instead of minor Ahmad princesses, which I think would be interesting, as it would make Danish nobility viable arrange partners for minor princes. We would likely see the Scheswig-Holstein minor dukes intermarry with the Danish nobility, which would serve to integrate these mire into the Danish state.
 
There wasn't a tradition in Denmark for Danish kings to marry common nobles, but in case he do marry her, the nobility will panic at first , but then they will decide, that it will make little difference. If Frederick and Anna produce children, it will mean a clearer succession than OTL regency council, which will be good news, if not Frederick have siblings/nephews to inherited him. It may create a tradition of Danish kings Harry Danish nobility once in a while instead of minor Ahmad princesses, which I think would be interesting, as it would make Danish nobility viable arrange partners for minor princes. We would likely see the Scheswig-Holstein minor dukes intermarry with the Danish nobility, which would serve to integrate these mire into the Danish state.

So there would be no decrying of the marriage as morganatic and Freddie deposed in favour of his uncle/brother? Always good news. I wonder though, wouldn't they try to prevent something like this (in their panic phase) from happening again? And would the marriage happen while Freddie is hereditary prince or once he's king? I can imagine it might cause ruptures in the immediate royal family, no?
 
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