Jim, personally I don't think that theme from the show IS predominantly ragtime, but swing. I'm hard pressed to think of ragtime that is really like it structurally or performance wise; it really is more like Glenn Miller/Benny Goodman swing to me distilled to a small group in a different instrumentation than say the small Goodman groups.
I think the Titanic disc will be interesting as an overview of pop music of the time; looking forward to it.
Jim, I'm not sure how conversant you are with early jazz in Chicago and elsewhere. Are you familiar with the Jimmie Noone sides with Earl Hines and Joe Poston? These have a flavor that may be somewhat in line with what you are seeking.