I don't know Warrack's source, but it will have been more than just 'Patrick Spens'; the term was a ballad commonplace, after all. As a commonplace, it may have no more contextual meaning than 'wee penknife' or 'milk-white hand' do; but the entry from Child's glossary bears selective quoting:

'braid (broad) letter ... either a letter on a broad sheet or a long letter ... [here he refers to examples where letters are 'lang' or 'large'] ... A braid letter has been interpreted to be an open one, a patent, but in almost every case here cited the letter is said to be sealed ... [further references].'

It may be that the term was little used except as a ballad convention, and there is no more to it than Jack suggests. Certainly any guess based on the coincidental similarity of the unrelated word 'braid' (as in plaited hair, ribbon etc) is not really a starter.