“Rain check” is from baseball. (Barnum did his things inside, so he’d have no need of one.)
Originally, when a ballgame was rained out, those who had already entered the stadium would go to the ticket office to get a refund. This was slow and, more importantly from the teams point of view, easy to cheat (anyone in the neighborhood – and ballparks were always in some neighborhood – could come by, stand on line, and claim to have paid for a ticket).
One club owner came up with the idea of returning part of a ticket, so fans could use those if the game was rained out. He called it a “rain check” and the name has move to include anything handed out to make up for a missing item at a later date.
Don’t know about ringmaster, though I suspect it was a circus term well before Barnum.