Lusitania
Donor
A Dutch Hudson Bay could become a beneficial revenue source for the Netherlands. After the loss of New Amsterdam and Fort Nassau, they owned no source of furs. Sourcing them directly from their own Hudson Bay colony would have saved a lot of money on importation of unprocessed furs, and allowed the Netherlands to profit by exporting manufactured clothing to Germany and Southern European countries. At the same time, the Hudson Bay would be a small enough segment of the total market (producing about a tenth of fur sales at the time) that the English wouldn't feel very threatened.
With additional production and a new area of investment, the decline of the Dutch empire would occur more gradually, although it would still probably decline due to competition with England. When industrialization begins in the 19th century, the Netherlands will have an easier time rising from whatever occurs in lieu of the French Revolutionary Wars, and they might end up returning to great power status.
As for the development of Hudson Bay itself, it would probably be much the same as under English rule, as its governance would be solely based on maximizing fur profits. Interactions with natives will be similar and the European settler population was never very high. The prairie provinces will be difficult to settle from Hudson Bay, and the territory will remain company ruled for some time.
Exactly, in 1815 lord Nelson tried to setup a Scottish colony but that failed due to long distance to get to habitable areas (Winnipeg) plus animosity between Metis and company administrators who wanted the area to stay based on fur trade not settler.
So the company faced with pressure from British colonies and USA would in the 2nd half of the 19th century sell either to britais Canada or USA or to both.
Unfortunately due to geography it could never function as regular colony. If the company did not sell it it would be over run by people from either of those two counties as prairies are discovered to be good for agriculture and resources are found. If dutchvtry to hold it they would suffer same fate the Mexicans did with California.