TL;DR: A Social Security number is not necessary if you are applying for the CRBA and US Passport at the same time
It looks like I need to provide a signed statement. I found this on the US Department of State website:
Do I have to provide by Social Security number on my passport application?
Yes. 22 U.S.C. 2714a and 22 C.F.R 51.60(f) requires you to provide your Social Security number, if you have one, when you apply for a U.S. passport or renewal of a U.S. passport.
What if I am applying for my child who does not have a social security number yet?
You will need to submit a statement, signed and dated, which includes the phrase, “I declare under penalty of perjury under the laws of the United States of America that the following is true and correct: (Child's full name) has never been issued a Social Security number by the Social Security Administration.”
I believe submitting the CRBA, Passport application, and this document, would get me out of the loop.
Update 2/21/2020 - After my visit to the US Embassy in Romania
The CRBA and Passport application were both approved! The declaration about the SSN was not necessary. I asked about the SSN situation and they said something along these lines:
Since you are applying for the US Passport at the same time as the CRBA, a Social Security number is not necessary. If you were to get just the CRBA now and then apply for a US Passport later, then a declaration would be necessary.
It seemed like they had these forms with the declarations on-hand at the embassy. In order to apply for a SSN, I either have to go (or mail original documents) to Naples, Italy, which would take about six months to get the SSN, or I can travel to the states and apply for the SSN there.