The issue in my mind is that the proposed "$3-a-head" pricing model is ambiguous. The first discrepancy is the number of 'heads' that were available to you. Customer promised one figure (high number) and you saw the DOWNSIDE of the available revenue potential.
Since this didn't work the first time, it's incumbent upon you to either: a) set a minimum cost number based on the highest number of children, i.e, $x per head with a minimum fee of $y; b) charge hourly for your services; c) pass on the opportunity.
Lastly, if you can agree to terms with this customer, put it in writing!