Denmark has a state church, which means that the church can influence politics, but also that politics can influence the church. So if church buildings are state buildings for marriage, and the state allows same-sex marriage, then those buildings have to accommodate same-sex marriages.

If the church has a problem with this, it should divorce the state.

In neighbouring Sweden, church and state separated in the year 2000, so when same-sex marriage was finally made legal in 2010, this was simply not an issue. The Church of Sweden can deny same-sex marriages on its premises and by its staff if it so chooses.

Separation of church and state gets a bit murky when there is a state religion.