I don't think i've ever seen anyone send in the blue passport for processing an official use brown passport. But that being said, check the foreign clearance guide for country specific requirements (you need to be on NIPR).
Wanted to give a follow up for what happened. I did go on the Tailswap, fantastic time btw! I took your advice and looked at the requirements for the countries we were traveling through or could possibly land due to diverting. All the countries we traveled through required a DoD ID card and NATO orders. Had any of the following countries we could possibly land at require a passport then I wouldn't have been able to go. We looked through the foreign clearance guide of all the countries to determine the entrance requirements. I'm sure some of you wouldn't have taken that risk but it worked out in my favor and am glad I got to go.