For correspondence between Twain and Barnum, you might want to track down
the items listed in Paul Machlis's _Union Catalog of Clemens Letters_ and
_Union Catalog of Letters to Clemens_ (both U of California Press).  There
are several hits for Barnum in both books.

There's an entry for Barnum in the _MT Encyclopedia_ (Garland, 1993),
which unfortunately doesn't tell much about his relationship with
Twain--but maybe the items listed in the bibliography of that entry will
lead to something more.

Hope this helps!

Taylor Roberts
University of British Columbia