This is the kind if project where agile can truly shine.

“Hey, customer, looks like we burnt through half your budget and it doesn’t look like we’ll be able to complete the full spec on what remains…

I could ask you to increase the budget, but I think I have a better suggestion. You tell me which feature is most important to get in and well focus on that exclusively, and iterate on that until you're either happy with what you’ve got, or the budget runs out. Then lets take a decision on how to proceed from there if still needed”