Ugh, what a horrible situation. We had a similar conundrum with an official passport for our daughter, which actually was processed and mailed to us two years after we applied. Fortunately, we did have a tourist passport for her so that was okay.
If your cheque cleared, might they waive the fee for the emergency passport? At least in part, perhaps? Might be worth asking - obviously the mistake is at their end. Just keep your record of the transaction! You may still have to pay a fee to have it expedited, but it's better than nothing.