How does a dentist treat baby bottle caries?

Depends. Hopefully this is caught before the situation is too serious. .. Fillings are the most conservative solution. If it is not caught it may require root canals and even extractions.
See below. It depends upon the severity. The child may need fillings, caps, and/or tooth extraction.
Eliminate infection. Fillings and/or root canal rx and/or extractions may be necessary. Please never put a baby to bed w a bottle containing anything other than water. Taking child to dentist, especially pediatric dental specialist, by age 1 could have prevented this heartache. Please don't let the infection progress...Have it treated now.