Danish conquest would not look like TO conquest and would be limited to coastal areas likely.
Seems like that in OTL the Danes managed to conquer only the Northern Estonia (Duchy of Estonia) and, because OP does not say a word about the Livonian Brothers of the Sword of Livonia, there is no reason to assume that situation in that region seriously changing comparing to OTL: Danes on the North and a mixture of the Church lands and Order’s lands in the rest of the territory (IIRC, the Livonian Order had been formally subordinated to Archbishop of Riga).
The obvious difference is that in an absence of the TO after defeat at the Battle of Saule in 1236 the Livonian Brothers of the Sword are not going to be incorporated into the TO and the alt-Prussian state is not getting a foothold in Livonia and various local events could go differently (in OTL the TO played main role in crushing the Estonian rebellion in the XIV century).
Nucleus of TO state was Chełmno Land, westernmost part of Duchy of Mazovia, and TO received substantial support from Polish dukes, something Danes would not receive.
Yes, while there were
initially some comprehensive reasons for inviting and supporting TO, the reasons for giving the task to the Danish state (even if formally separate from Denmark) are much less clear and the schema looks more dangerous for the Masovian/Polish side. After all, the TO was a reputable religious institution and nobody could guess that it is going to grow into a reasonably powerful state but trusting conquest of Prussia to a foreign duke would be just an invitation for such a state to get created.
BTW, just out of curiosity, would an invited Danish princeling have military resources needed for conquest of Prussia p his own?
Also, various monarchs, including Czech king Premysl Otakar, after whom Konigsberg was named, came with their forces to support Monastic State. Czechs would not bother to fight for Danish king or prince.
Yes, the monastic “component” made a continued “crusading” effort popular in Europe all the way to England and France (Henry Bolingbroke, Boucicaut and less prominent figures) but helping some obscure duke to fight the local heathens does not look equally attractive.
Also, even OTL conquest of Prussia by TO could collapse due to Herkus Monte's uprising. Prussians were tough nut to crack and Prussia was more inland and more populous than Estonia (and more exposed to Lithuanian raids).
It took TO fifty years to conquer Prussia and for sustaining a needed effort the secular duchy would have a considerable amount of its own resources besides outside help. The TO had such resources but the dukes would be completely dependent upon a continued help from Denmark. Tyehen, TO deported or killed a considerable percentage of the local population and compensated by sponsoring immigration from the HRE and Masovia. Did Denmark have a surplus of the population big enough? Wouldn’t it be rather dangerous for a Danish ruler to have mostly foreigners (including nobility) as the subjects?