AH
I am afraid that the criteria for IB are quite different from those for ESA, so it is unfortunately possible to be properly awarded IB and to not qualify for ESA.
I am sure you have already looked, but our ESA MR & appeal guide is on the following link, which also has template letters that you can use for requesting your MR.
www.benefitsandwork.co.uk/help-for-claimants/esa1 If you have any questions about the process please come back to the forum.
Gordon