OP - I would stop focussing on your buyer's financial situation. To be honest if he wanted to buy your house for his budgerigar it doesn't affect you. Him wanting to save the stamp duty is normal and people all over the country will be doing it.
I can see that added to the normal stress of moving house the fact you are going to a house and area you don't like as much as you have now is not enthralling. However, if the reasons you did it (i.e. compromising on those two things for a few years for the children's schools ) were right when you made the decision, then they are right now. If so, then having secured a house in the small catchment area is valuable to you. If the catchment is small and you pull out now, what are the chances of finding another house , completed and moved into in time for October?
What you have now is much lower risk, and very possibly worth having paid to expedite surveys etc.
I don't understand why you would be liable for his stamp duty. It would surely be the case that you and he and your vendor get your finances lined up and exchange and complete only when all 3 are in place. If that doesn't happen then the risk is he decides to pull out and look again at leisure. And how does that then affect your purchase? Very often the complaint is that the buyer is dragging their heels - at least you don't have that.
I agree that you do not know if your house is undervalued. You said you got a "decent market price" . If you feel you are paying "over the odds" for the new house - might that have something to do with the catchment. In which case you will (in theory) benefit when you come to sell that house.
Your problem I agree is the removals but the insurance a pp mentioned above sounds the way to go.
It will be a nail biting few days but then you will have 5 months to get the house sorted and settle in before school starts. I would rather have a weeks stress now than start again with no guarantee it won't all happen until October.
Not sure how helpful that is but just advising to concentrate on your priorities and how near you are to having achieved them. In a week's time you will very likely have achieved all you set out to do.
Easy for me I know but just trying to remind you of the positive 