Can you charge for the offer based upon a percentage of what you will save the taxpayer?

No. Circular 230, section 10.27 states that someone practicing before the IRS cannot take matters on a contingency basis except in a few limited circumstances, including:
A. An IRS examination of an original tax return or an amended return or claim for refund or credit,
B. Services rendered in connection with a claim for credit or refund filed solely in connection with the determination of statutory interest or penalties assessed by the Internal Revenue Service, and
C. Services rendered in connection with any judicial proceeding arising under the Internal Revenue Code.
Collection matters are not one of the enumerated exceptions, and hence cannot be taken on a contingency arrangement. Beyond the ethical issue, it never made sense to me that a practitioner would take these matters on contingency because there is little chance of being paid later. Most of these clients are doing Offers-in-Compromise because they cannot pay the taxes. I think it is unlikely they would be expected to pay a percentage of the tax savings afterward.