I have recently updated kicad to v6.02 in linux mint 20.3
Have configured Schematic Editor to add manf# in “Field Name Templates”
When try generate BOM the default installed is “bom_csv_grouped_by_value” and “bom_csv_grouped_by_value_with_fp”.
When open the CSV file doesn’t appear the column with manf#.
Not an easy way, you will need to modify python bom script to your taste (write your own, with reference to preshipped ones). I am using my own highly customized script long time ago, so don’t have a clue if any easier method is available now. Pleae see this discussion: BOM Script: a way to exclude part from BOM by custom field