I agree with EVERYONE else. Refund and DO NOT engage with this customer. You contradict your own statements when you say "I need the money by XXX date or I just can't do it" then they pass said date and time and pay you and miraculously you can now do it. For some reason some people think they are doing US the favor with an exceptionally under priced order...enjoy your weekend instead!
Don't move your line in the sand. Red Flags are there for a reason...don't ignore them. If you want your customers to take you seriously then you have to stick to your guns. It is, after all, THEIR loss. Refund it. imho.....
Cake disasters are like childbirth...you swear you'll NEVER make another cake but soon the pain is forgotten and you are elbow deep in BC again.
What are they thinking? You tell them the cake has been cancelled so they pay you? That saying "you snooze, you loose" is definitely applicable here. They are trying to guilt you into it because "I paid." I agree with everyone that you should send the money back and not make the cake. Your are setting yourself up for trouble.