"There's one problem I have though - how do you sing a song containing Scottish vocabulary without slipping into a phoney accent to match it?"

A very important and sensible question.
1) If there is just an odd Lallans word in the song – sing it without an assumed accent and don't worry.
2) If there are a few such, see if you can replace all/most of them with standard English equivalents. If so sing it without an assumed accent and don't worry.
3) If there is a one off reason for singing it (the writer died the day before, or your Scottish grandmother has come to watch you and desperately wants you to sing it) fine – apologise, explain and do it with cod accent if you need it. Once.
4) Otherwise DON'T SING IT. Is that harsh of me? It's just that there are many many thousands of folk songs out there, most of which do not require an assumed accent, and which would be perfectly good alternatives. Why do people have to insist on their right to sing exactly what they want to sing, irrespective of whether by any objective standards it is a good idea?
I see exactly the same thing happening with the 'ownership' of songs. At one time (and I don't just mean in Traditional Land; it was the case in most folk clubs until comparatively recently) if a singer in a particular area was known to sing a specific song, noone else would sing it when that person was there. In many cases not even when they were absent. God forgive me, I can remember in the seventies waiting anxiously for a certain person to carry out his promise to move to another area so I could lay claim to 'Harry Was A Bolshie'. Nowadays I've had people ask me for the words of a song (which as always I have willingly given) and within a few weeks they've been singing it there in front of me.