The story most likely did not originate in Denmark at all, since Jews in Denmark never wore the yellow star. So the story wouldn't make any sense as rumor or propaganda, at least not for a Danish audience.

During the first part of the German occupation, the Danish government was left in charge, as long as they cooperated. Which they did "under protest". No discrimination against Jews was enacted in this period. In 1943 the cooperation broke down, the Danish government resigned, and the German forces took over governing. Shortly after orders were issued to round up the Jews in Denmark and send them to concentration camps. (Most of them escaped to Sweden though.) So persecution of Jews in Denmark went from 0 to 100 in a short timespan, without the intermediate phases known from other countries with gradual erosion of civil rights, markings with yellow stars, isolation in ghettoes an so on.

The king Christian X did become a national symbol of sorts during the occupation, but his most brave act of defiance was sending a relatively terse thank-you note to Hitler, in response to Hitlers gratulation telegram on the kings 70 year birthday. This terseness of the telegram was considered a slight and caused a minor diplomatic crisis.

The legend of the king wearing a yellow star apparently stated with a cartoon in a Swedish newspaper in 1942. http://scandinavianjewish.blogspot.dk/2016/09/ragnvald-blix-born-in-oslo-son-of-e.html

In the cartoon, the Danish prime minister Stauning is conversing with the King.

Stauning: What should we do, your majesty, if Scavenius* enforces that also our Jews should wear yellow stars?

The King: Then I suppose we will all have to wear yellow stars.

So this is a fictional conversation about a hypothetical which never happened.

(*Scavenius was the Danish foreign minister at the time and primary liaison to the German government. He was and is a very controversial figure since he had the thankless task of ensuring the Germans did not feel the need to install their own government, as they had done in Norway. But many felt Scavenius went too far in appeasing the Germans.)

The linked page have more details about how this story developed into the myth you know. Apparently it was mostly spread among exiled Danes in the US and in American media, which makes sense since Americans presumably would be less aware of the differences in how Jews were treated in the different occupied countries, and the yellow start was a well-known symbol of the persecution of Jews in Germany and occupied countries.