Apart from the Proclaimers, all the Scottish rock bands sing with American accents and generally get away with it. I don't see why north Americans shouldn't be able to reciprocate.
I would think, however, that to sing in Scots dialect, you would need to spend just a wee bit of time learning, not only the words, but the meaning of them so that the song would come over with more sincerity.
I recently got a CD from Mary Smith (marynf)of Virginia and she does a really good versions of "Glenlogie"
When I listen I don't hear an accent but just someone singing a ballad well.

I would love, however, to hear from across the pond if you have ever heard someone from this side murdering an American ballad.
I've heard Hank Williams songs being sung with very broad Glasgow, Inverness and Peterhead accents. It all adds to the colour (color) of life.