Had to look in crown douche was from the same nation they reside in. Although the spelling is different, it seems more like a translation issue, so he would technically have the right to do so, if it weren't for most laws in a kingdom.
For instance, being the judge, jury, and executioner, as well as possibly invading a foreign nation for a faux witch hunt. Unless the concept doesn't exist in this world, I could almost guarantee that the og kingdom would secretly register that mansion as an embassy with the higher ups in the new kingdom. One of the key details of all embassies is that the new kingdom's rules do not apply in that property, and the property follows the laws of the og kingdom.
Three more things I want to ask: 1. If a lot of people have magic, what is a witch? If it is non-human, wouldn't that exonerate the main characters? 2. If the kingdom is run by seven merchant families, why is there a crown prince? 3. If the crown prince even did follow through with the execution, once those three villages saw what happened, word will get out and the kingdom will take a beating from both foreign forces, and domestic.
To summarize it all, crown prince is an idiot with the foresight of a naked mole rat, and that haircut of his makes the more rat look attractive