Hey folks - I wanted to share an update here so that anyone facing a similar problem knows how this worked in our specific case if facing a similar one in the future.
My husband was able to leave the country on his German passport, then successfully submit a passport application using form DS-11 with his Naturalization Certificate as proof of citizenship at the US Embassy in Berlin. He'll receive his passport at the German address we're staying at in about three weeks time.
As other users advise above, I'd encourage you all not to violate a federal statute, especially not if other options work for you. In our case, our Senator said "Can't help you," and there were no emergency or urgent passport appointments available anywhere in the US, even in Hawai'i before my husband and I needed to be in Germany to take care of his father. Still, if you have no other option, it is an option to leave on one passport and apply for your first full passport abroad as a naturalized citizen.
For anyone curious about more detail: The embassy was not thrilled with my husband when he first arrived to his passport application appointment and the first question he was asked was, "So why did you leave the US without a passport? This is not the way to do this." Upon explaining the situation, the case worker gruffly responded that, "Well this isn't a real passport we give you, it's an emergency one-use passport." My husband asked if a real one was possible since he was going to be in Germany taking care of his father for two months and the case worker's mood immediately changed. After that, it was no problem at all and the entire appointment took about ten minutes with a much nicer tone to the conversation. My takeaway is that the case workers do not like having to issue emergency passports to non-passport holders, but are more than willing to issue first passports to citizens outside of the country if they have time to wait for a passport. The case worker quite bluntly acknowledged that the passport situation in the States was absurd.
Hope this helps others in the future!