Hey Alura,
First off all congratulations! What an exciting time!
You have 17 years to sort out an N-600K, so I recommend that you try the CRBA with the Consulate first.
If that is not successful then come back here and let’s talk.
To answer your question though, the purpose of the N-600K is to get a Certificate of Citizenship to prove your child is a citizen. It can then be used to get the US passport. So yes, you can get an N-600K.
The downside of the N-600K vs CRBA is the expense ($1000+ USD) and time taken. The N-600K can take a few years.
Try the CRBA first.