As a USN, you may not be denied entry to the US. Period. Even if you don't have a passport at all. Nonetheless, US law requires that when you enter the US, you enter on your US passport. It is a violation not to. But, you can't be denied entry, you may simply have a fine imposed (less than a remote possibility on your facts). Thus, in the final analysis you will get to your treatment and should stop worrying.

Nonetheless, the greatest hurdle is being permitted to board the aircraft.

For this reason, I suggest calling the Consulate. They ought to be able to renew your passport on the spot based on your proof of travel arrangements and the reason or at a minimum, provide a temporary (passport replacing) document which permits one-entry within X days (allowing your passport to catch up with you in the US).

Given that you have other stresses in life and things to concern you now, rather than testing the limits of the system, simply call the Consulate, explain the situation and ask for help. It won't be a problem, it's what they are there for and your situation is not that unusual.