Obviously, there had to be some kind of agreement. If it was not written, then the practical problem is proving the terms of the contract. It is not reasonable to wait for two years to send you a bill, but I do not see how that would have prejudiced YOU. You are not paying more than you would have, because of the delay.
I do not understand why the bill would be excessive, if you agreed to the earlier terms. If there was no ultimate price term agreed to, then that would be a different story.
The bottom line is, if the work was done, then you owe them for what was done. There would be undue enrichment if you were allowed to enjoy the benefits of the agreement without having to compensate the contractor for the work. The leverage you have is that the work was done and you have the money in your pocket. In order to get the money if you are not cooperative, the contractor would need to take you to court.
In order to avoid taking that route, they may be willing to negotiate with you. But I do not think you have any grounds to force them to accept less, merely because they were late in invoicing you. You have benefited from their tardiness by being able to keep your money and invest it, while they were out of pocket for work and materials. Absent some sort of discrepancy in what they are charging versus what you agreed to, I think you should simply pay them.
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If the bill is clearly excessive call and complain... late submission will look bad if you ultimely refuse to pay and they have to go to court to enforce it. So if you offer them immediate payment of part and they agree to a settlement they may accept.
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