The only printed version of the song that I'm finding is from here:
https://archive.org/stream/laterpoemsofalex00ande/laterpoemsofalex00ande_djvu.txt
This is a 1913 book titled Later poems of Alexander Anderson, "Surfaceman". So, are we to take it that this song was written by Alexander Anderson (1845-1909)?

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Thread #158499   Message #3749271
Posted By: GUEST,#
08-Nov-15 - 11:47 AM
Thread Name: Origin: Bonnie Bessie Logan
Subject: ADD Version: Bonnie Bessie Logan

From pp 115-6 at

https://archive.org/stream/laterpoemsofalex00ande/laterpoemsofalex00ande_djvu.txt

BONNIE BESSIE LOGAN.

O, BONNIE Bessie Logan
    Is dainty, young, and fair;
The very wind that's blawin'.
    It lingers in her hair.
Sae lichtsome is her footstep
    As she comes o'er the lea ;
But bonnie Bessie Logan
    Is owre young for me.

O, bonnie Bessie Logan,
    The lads are at the stile,
Or half-way up the loanin'
    To catch your winsome smile;
I fain wad be amang them,
    If sic a thing could be,
But bonnie Bessie Logan
    Is owre young for me.

O, bonnie Bessie Logan,
    I saw you late yestreen ;
A rose was on your bosom,
    And love was in your een.
I doot the lad that pu'd it
    Is sure to win his plea,
For bonnie Bessie Logan
    Is owre young for me.