I believe the name Tircuit is of French origin. However, the particular ancestor that you're looking for most likely has Spanish ancestry also from another side in his family. Same is true for my ancestors, ex. Pierre/Pedro. Many Spaniards living in Louisiana changed the first name to the French form.
You said that Francois/Francisco Tircuit's wife was Maria Barbe Bourgeouis. Her also, probably French-Spanish mixed ancestry from her first Spanish name. But there again her last name is of French descent. Who would have to research their lineage to find further details.
I hope that was helpful.