Look at Burns's letters. The link is as certain as anybody could want:

https://en.wikipedia.org/wiki/Scots_Wha_Hae

He wrote the song within a day or two of Muir being landed in Scotland, a few miles from where he was at the time. I'd assumed he actually saw Muir, but from what Jim has dug up it seems he just missed him.

I have some stuff in my "Embro, Embro" pages about that period. The biggie was the King's Birthday riot in (Edinburgh's) George Square in 1792, which was directed more against the slave trade than anything else. The most explicit verse/song statement from the time is "The Tree of Liberty", which is often attributed to Burns, though I think it's equally likely that James Tytler wrote it.