I found several sources that agree that the spelling of the lyricist's name is Thomson, not Thompson. They also agree that the spelling in the title is Robbie, not Rabbie.

Here's a profile of the author in The Caledonian, Volume 5, December, 1905, page 20.

A profile of the village of Bowden, which was Thomson's birthplace, also mentions him and the song, in The Border Magazine, January, 1907, page 24.

The G. Ross Roy Collection of Robert Burns: An Illustrated Catalogue gives this description of the sheet music:

The Star o' Robbie Burns: Song. Words by James Thomson, music composed by James Booth. Glasgow: Mozart Allan, [1950-1969?]. 1 score (7 p.).…