IMO tell that builder where to go thats 1/1.5 days work tops to sort that.

If i was you i would rip down the current ceiling and have a go at reboarding it you'r self its not that hard. If you do go this route buy 6"x3" boards as there easy'r to handle than 8"x4" it may take you a day or so doing it yourself but as you say the money will come in handy for other things

Then get a plasterer in to give it a skim