The first ticket is yyz-ord-hnd-sin so the baggage transfer before then shouldn't have been a problem (you don't have to pick up your bags in ORD coming from Canada as you go through customs and security in Toronto). What I wanted them to do was to add SIN-CGK to the end of the trip on a Star Alliance carrier (Singapore Air)--although they were not interested in the airline, just that it couldn't be done. FYI I was very polite during all of this and didn't raise a fuss at all.

So any advice on what I should do to get my bags with me to CGK? I'm still in the lounge in Toronto.