Thanks for that, Matt, and I'm glad you like the site.
I agree that that The Hammond School's treatment of Gallows Child worked well. Partly, I think because it was for a Christmas show anyway (which made the tune doubly appropriate), and partly because it has exactly the same tone as Dickens' own more sentimental passages.
On the question of the printed sheet's sometime-clunky lyrics, I've already said to a couple of potential contributors that they should feel free to tidy up the words where that's needed to make them performable. The guys who wrote these things were not remotely precious about their work, and I'm sure much of it could benefit from the same folk process routinely applied to old ballads we learn from hearing them sung.
On that subject, my new Pretty Polly essay has lyrics for an 1850 music hall parody of The Gosport Tragedy, which mocks just the forced rhymes you mention on the old ballad sheets. Spelt out phonetically to mimic Sam Cowell's on-stage delivery, it opens with the lines: "Young William, he courted her to be his dear / And he by his trade was a ship's carpen - tier."