Request for Equitable Adjustment (REA) - invoicing on a T&M contract

We have a hybrid contract that is Fixed Price (FP) and Time and Materials (T&M).  The government approved our REA and executed a contract modification that increased the contract ceiling and funding to include the REA amounts by CLIN (FP and T&M).  The REA calculation for the T&M CLINs was based on the original PROPOSED level of effort (LOE). 

We submitted a true-up invoice related to the REA.  The true-up invoice included the ACTUAL T&M hours incurred.  The ACTUAL hours were less than the PROPOSED hours.  As a result, the invoiced amounts for the T&M CLINs were less than the amounts on the contract modification for the T&M CLINs. 

The government came back to us and instructed us to adjust our invoiced amount on the T&M CLINs to match the executed contract modification.  Meaning...the government is asking us to invoice MORE than which we think we are entitled on the T&M CLINs.

Do you have any guidance that would allow us to invoice a fixed amount for an REA on a T&M CLIN?

Example:

CLIN 1 was T&M.  Originally Proposed 100,000 hours at $20.00/hour = $2,000,000.  The REA included 100,000 hours at $22.00/hour = $2,200,000.  This resulted in a contract modification adding $200,000 of additional ceiling and funding.  The actual hours incurred and invoiced are 95,000.  We submitted a true-up invoice to the government for 95,000 hours at $2.00/hour = $190,000.  The government came back and told us to invoice $200,000 to match the contract modification.  In my eyes, this would result in overbilling the government (tax payers) $10,000.  Our contract is much larger than this example and the 'overbilling' amount would be ~$600k.  If there is any way we can justify billing the higher amount, I would love to hear thoughts.