If the communications in your email are clear enough, then it does constitute an agreement/contract between the 2 parties.
If you have done work for him in the past under T & M, Cost plus, PM type of contract or if you advertise that you do then he does have an argument in assumptions.
You then have to prove that a fixed price was negotiated/accepted.
Do you have documentation of the work that you performed? Pictures?
Either ways, you got yourself in a pickle. You gave him grounds for an argument. To him, it looks like you charged $1250/day for 2 guys. $625/day for 1 guy. $78/hr each. He probably saw that and got heartburn.
I agree, you should have never opened up the books for him in any way and stuck with the principal of fixed price means fixed price. He's not an idiot...an accountant of ALL PEOPLE! He broke down your price before he was finished reading your email...you didn't fool anyone with your skewed math.
I'm not saying don't charge $78/hr...I'm saying if you do and you feel the customer can't stomach it, then don't show them. AND CERTAINLY don't play around with the numbers trying to throw them off. You would have been far better off by standing proud behind your hourly price.
That's how you lose trust.
Problem is, if you back down now then you lose all credibility. Stand behind your price, the fact that it was fixed, you performed the work, went as far as doing the unprecedented and opened up the books for him and NOW HE HAS TO PAY YOU THE FULL AMOUNT. Done, end of deal, now or I slap a lien on your property. Turn it around on him and tell him that you forewent the signing of the contract because he is a past customer and you trusted him and that now he has further betrayed your trust. Let him feel bad, not you. Present your argument of how unethical HE IS!
You could legally charge $10,000 a jug