Who should sign COE extension request first?

I encountered an issue in escrow with a Fannie Mae listing.  Fannie Mae's attorney needed to extend COE on the basis that they were unable to sign off on HUD in a 48 hour time period.  My buyer had already signed docs and the lender was ready to fund.  The listing agent wrote up the extension amendment on a Fannie amendment form and asked me to have my buyer sign the extension requested by seller without having the seller actually sign their own request first.  I let them know that the seller/Fannie Mae needs to sign their COE extension amendment first and the listing agent responded by saying the seller will NOT sign first due to Fannie Mae protocol?  She claimed that if my buyer did not sign  the extension then due to the fact we would be out of escrow -the "computer system" would automatically kick the deal out and the entire process would have to start over at that point.  She then stated that nobody involved wished for that.

Is it better just to have buyer sign the sellers extension request or risk the losing the transaction in its entirety due to a technical issue of who needs to sign what first and when?

I'll let you be the judge and decide if the transaction closed and on what terms..