It's not a complete answer, as the 15 coins could also be 9 dimes, 1 nickel, and 5 pennies.
I think you're on the right track, but the final solution must factor in all the possible combinations of coins that meet the 19 and 15-coin criteria.
I see from BMAD's earlier respose that this is his intended interpretation, but to me the wording of the puzzle doesn't make that clear. As stated, the puzzle seems to have the bartender blindfolded and engaged in a minimum of one "round" of turning over either 1 or 2 glasses, as only "After each round, the bartender is told if all glasses are oriented the same and the game is over.".
Fun puzzle though either way -- and the logic in plasmid's solution extends to either interpretation.
Thanks to all.