My apologies if the link I (attempt to) provide in this post
has been posted already,
elsewhere on the Mudcat forum.

This is Ron Radosh, who triggered the whole "Big Joe Blues" thing.
First he wrote an editorial in light of
a recently released Pete Seeger documentary,
asking Seeger to "repent" as regards Stalin.

Then Seeger sent him a letter, which Radosh details in
New York Sun, August 31, 2007.

Radosh's follow-through editorial reports that
"Mr. Seeger attached the words and music for a song he had written"
to Seeger's letter to Radosh.

Which, I guess, is the closest thing
I'm going to get to an answer to my question.
The answer being:
yes,
"The Big Joe Blues"
does have music.