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We do have a "non-binding" estimate, but they can't turn the wheels in their large organization to get payment for more than the high end of our non-binding estimate. And no one takes us seriously if we say "probably 10 to 12 hours, but this kind of work occasionally has problems that will balloon the costs to 20 hours."

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Why would you not just say 10-20 hours then? If you know that you're going to wear any overrun costs, you owe it to yourself to give the estimate based on the worst case, NOT the average case.

Edit: If you think that 20 is a real outlier, and 12 is your average case, use something like the PERT formula: ( [Optimistic] + [4* Realistic] + [Pessimistic] ) / 6

If O=10, R=12 and P=20, that would give you an estimate of 13 hours.

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