I have been trying to find out more about "BUI" as a Last or Surname.
If you use the Free Name Search on the Home Page of Scotland's People, the significant results are back in the OPR period, prior to 1855.
There are 117 BAPTISMS or Births in that total of about 300 OPR Years.
I suggest that you use some of your SP Credits, to look at those Search Index Results, to get some idea of the location/s involved. If concentrated in or around Jura, it suggests that the name may be based on a locality in that area.
I have found that BAPTISM or Birth in the OPRs; BUT I had to drop "Finvall" from the mother's names to locate it. Does that name "Finvall" actually appear on the Scanned IMAGE of the OPR?
There is one occurrence of "Bui" in the Wills and Testaments, for Dunblane.