In the 16th century, Danish king Christian IV required captains of ships passing by Denmark to pay taxes.

That's correct, but a bit misleading the toll started in 1429 and went until 1857 with the specifics changing a bit over the period.

They were allowed to declare their cargo at whatever value they chose, without further audit, but the king reserved the right to buy the cargo at that price.

This is again somewhat correct for most ships but many where also audited and their cargo audited. Some goods had set tolls others where by value, when it went by value that was the method employed for determining the toll, typically as a % of the value for a type of good. In 1753 about 33% of the ships where inspected and audited.

and were there any cargoes which were in fact bought for the declared price?

Yes though relatively rare, but there are still many examples, the ledgers that have survived on it are extensive at about a quarter million pages, but a notable example of cargo being bought could be a green parrot bought in 1534. As far as I'm aware no one has yet counted how often the cargo was bought.

The ledgers are currently available at: http://www.soundtoll.nl/index.php/en/over-het-project/sonttol-registers

But actually reading them is less than easy.