Hi Geoff,
In the early 1800s the nearest Catholic Church to Closeburn was St Andrew's in Dumfries town. That probably doesn't help because it wasn't built until 1815 when it became the Cathedral Church for the whole of Galloway the Diocese. It was destroyed by a fire in the 1960s, but the Cathedral then moved to Ayr.
Prior to the 1815 date the Maxwell Herries family, who held a lot of land in the area, retained a priest in their large estate house in Terregles. The House no longer exists, the estate was bought up by Government after WWI and broken up into smaller farms.
Some time ago I recall looking at Catholic baptisms etc in a Register from that pre 1815 period held by the Dumfries Archives Centre. These archives are now relocated with the Ewart Library in Dumfries.
That is all I can tell you, I suggest you contact the Ewart Library in Dumfries who may be able help with your specific enquiries.
Cheers,
Sy