The stanza in question that describes the dying of the petticoat and some of the others that Shule Agra and Lichtbobs Lassie have in common are 'commonplaces' that occur in several old songs. An eighteenth century version of that same stanza runs thus. 'The Hairy Cap'/'Jack and his Dandy Hat' of which there are several versions.

Then I'll go and sell my gown you know,
Likewise my scarlet roccelo,
I'll sell them upon my word,
All for to buy a good sword:
I'll look as rakish as young Jack,
With a scarlet cloak and a Hairy Cap.

In an English oral copy
'Sweet William among the Rushes' it runs

I'll have my petticoat bound wi' red (dye in the slip song)
And the lad I love, I'll beg his bread,
And then my parents'll wish him dead.
Sweet William in the morning among the rushes.

This derives from a late 18thc Yorkshire slip song 'Sweet William'.

In Herd 1776 we have

I'll sell my rock, my reel and tow
My gude grey m,are and hacket cow,
To buy my love a tartan plaid,
Because he is a roving blade.

In beaumont and Fletcher's 'The Two Noble Kinsmen' 1634 we have possibly related stanzas with other colours brought in.

For I'll cut my green coat a foot above my knee
And I'll clip my yellow locks an inch below mine ee
Hey, nonny, nonny, nonny.

He'll buy me a white cut, forth for to ride,
And I'll go seek him through the world that is so wide.
Hey nonny etc.

FWIW EmmaB seems to know what she's talking about.

The meaning of the red cloth could easily change from century to century as the verse has been reused in different songs, but, Jonathan, I presume your slighting remarks on those who are trying to put some historical background on this, are tongue in cheek!