I wouldn't pay the difference either.
That quote was obviously a pretty big factor in your decision on whether or not to go ahead with the work. Would you have gotten the work done for $6k? Who knows, but that's not your problem.
They gave you a quote, you agreed, don't feel guilty about holding them to their word (even if it was mistaken), because they woulnd't feel any guilt about taking money you didn't intend/can't spend.
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