It all sounds rather unlikely. If Frederik VII had indeed produced a legitimate daughter, the Schleswig-Holstein question would have been complicated yet further.
Yes, and some would see that as proof of why she was whisked away.
however, Frederik's third marriage was morganatic, so unless he was able to alter the legal position of any offspring following their births, the children of that marriage could not inherit in any event.
It was debatable wether Danish law, unlike the German law applicable in the Duchies, recognized morganatic marriage as anything else than a testamentary instrument used in connection with the monarch's bigamy.
Two earlier barren marriages anyway suggests that the inability lay with him, rather than his wives.
Is it possible that the child was an embarrasment because the actual father was Countess Danner's former lover Berling and it would be a pity of there was a succession crisis / war because of a child whose biological father was not the king?
and finally, until Danish law changed in the 1950s when it became clear that Fred3erik IX would only have daughters, no woman could succeed to the Danish throne.
Strict agnatic / Salic succession only applied from 1853 to 1953. Before 1853 (the child allegedly being born in 1851) the succession was regulated by the Lex Regia, which had agnatic-cognatic succession. With all the male lines exctint, the next successor was Landgravine Louise Charlotte of Hesse-Cassel née Princess of Denmark. To keep the agnatic-cognatic succession in the Kingdom tied to the agnatic succession in the Duchies, the Succession Law of 1853 settled the (henceforth agnatic) succession on her daughter's husband, Christian of Glücksburg and their agnatic issue. But originally the Lex Regia had ordained that the succession was not just agnatic-cognatic, but that the female herself, and not her husband, was to succeed:XXXVII. Det er Døttrene og døttrenes Børn og Børnebørn i ævindelig rad, som Arffvesuccessionen udj Regieringen skall tillhøre, og icke døttrenes Mænd, som aldeelis intet med disse Rigers Eenevoldsherskab skulle sig befatte, og i huor høye og megtige de hellers kunde være i deris eygne Lande, skulle de dog, naar de herudj Riget ere, alldeelis ingen magt sig tileygne, mens altid ære Arffvedronningen og hende den høyre haand og ypperste sted giffve.
XXXVII. It is the Daughters and the daughters' Children and Grandchildren in eternal line, to whom the Hereditary Succession shall belong, and not the daughters' Husbands, who nought must have to do with the Absolutist Government of these Realms, notwithstanding how high and mighty they may be in their own Lands, they must though, when they are here in this Realm, no power whatsoever acquire, but always honour the Hereditary Queen and give her the right hand and superior place.
One can start to dream up conspiracy theories: Did the succession Law of 1853 come about precicely because of this child? Those evil Glüksburgs.... did their crimes not just start with scheming to get back Schleswig, thereby causing WW1 and the Russian Revolution, but actually with depriving "that woman" Rasmussen's child of her inheritance?