I don't believe BOM tables retain their template name / location information. I can't seem to find any reference to that in the API commands.
However, if you see a BOM table with a format that you like, you can right click on it and do "Save As" to save the template as a .sldbomtbt file.
As you write, I can "Save As" table, but I save it via macro (I need other format, some other rows, collumns etc...) but macro will works only for one bom template. It will be good inform users about wrong bom template.
However, I could "copy" table into excel, and chceck data (each template has other format) and if it's wrog, that Solidworks finish Excel and macro. Excel will works "in solidworks."