Only one authority should be cited on a modification document because it is the part that indicates what allows the modification action in the first place. In your example, commercial changes are a bilateral modification action, requiring both parties to sign (acknowledging agreement) that something of the terms and conditions identified in block 14 has changed. Exercising an option is a unilateral modification, so you wouldn't require the Contractor's acknowledgement with a signature and is contingent on proper advanced notification to the Contractor that the Government intends to exercise the option.
Also consider that when you execute a modification with two different authorities, how are you going to report that action to the Federal Procurement Data System - Next Generation (FPDS), where only one action can be selected? If your modification invokes two different authorities, let alone two different modification types, then how are you reporting accurately to FPDS about your modification? This single-selection FPDS feature implies that there is only one contract authority per one modification action.
And since it was mentioned in the thread, utilizing “mutual agreement of the parties” should only be used as a last resort in which the circumstances aren’t supported by a FAR authority (i.e. Contract Clause). I am attaching a useful guide on utilizing block 13 for contract modifications to share with everyone here. I understand that there are various practices around the workforce, but consider what are the best practices, and understand why it is.
Contract Modification Authority Guidance Chart (SF30, Block 13).pdf