What if the IRS calculates a higher amount than the amount my client offered? Is the Offer dead?

No, the Offer is not dead. The IRS will generally propose the higher amount and if the client accepts, then they would be required to sign an Addendum agreeing to increase their Offer, and make a payment to bring the amount paid in to the IRS in-line with the amount Offered.
For instance, if the Taxpayer had Offered $5,000 as a lump-sum to settle the tax debt, they would have paid in 20%, or $1,000, with the Offer when it was filed. If later the IRS Offer Specialist determines they would propose acceptance at $7,500, and the client agrees, the client would sign an Addendum (Form 14640) and make a payment of $500. The reason for the $500 is that for a $7,500 Offer the taxpayer would have paid in 20% with the Offer, or $1,500. The taxpayer had paid in $1,000, so now must pay in the additional $500 to bring the amount paid in to the 20% needed for a $7,500 Offer.