A regular wedding on the Dutch side is civil ceremony usually followed by a religious one. The civil one is the required one. It has a lot of paper work and needs translation. Renewing of vows are much different and doesn't require any paperwork to do it.
I understand what you are saying about a JP or an equivalent and also about the religious denomination preference. Not sure if you could persuade a civil servant to do this without a lot of red tape and paperwork without local connections.
I'm sure this is done a lot on the island as we did years ago but sometimes not as easy as it should be.