FriedrichS

As per travel state website: "If you are requesting a transfer for a K Visa, the receiving Embassy or Consulate has the discretion to approve or deny the acceptance of a K visa application from an applicant outside the consular district.  Contact the potential gaining U.S. embassy or consulate in writing to request a transfer of your case and include the reason for the transfer request.  If you paid the MRV fee at the original post and the transfer request to a new post is subsequently approved, a new visa fee will be required."
 
So you need to choose where you can travel to process the visa and then contact them to see if they accept your case. 
 
Regarding the passport, many countries don't have biometric passports yet or began issuing them quite recently, so lots of applicants worldwide get their visas in their non biometric passports. 

Never heard of non-biometric passports were not good enough for a US Embassy. My daughter when she was a minor used to have one and the US Embassy in Moscow had no objections to sticking a visa in her passport. As long as a document issued by the state authorities it should be accepted by an Embassy. Your fiancée should apply for non-biometric passport for the daughter asap and get out of the country.

I can't speak for every embassy, but when I contacted Belgrade and asked about biometric passports I was told they were not a requirement as long as the passport was machine readable. The ones Russia is currently issuing should be fine for a K-1.