I can think of children who fit both of the possibilities that you suggest. That is, sometimes treatment effects generalise from the first to the second language, and sometimes not. When working with bilingual children, I start treatment in the language that the child and parent speak at home, and ask the parent to collect daily severity ratings for both languages. These severity ratings become the basis of our discussions and decisions about whether or not to treat in the second language.
To begin with, treatment in the Lidcombe Program is delivered by parents, not speech-language pathologists. It seems a minor point, but if you’re clear about that, then the rest follows easily. The mother treats this girl every day, so they can use the language they prefer – which seems to be Arabic. Your role is to train the mother to take speech measures, and correctly and safely present verbal contingencies following the girl’s stutter-free and stuttered speech. I have lots of experience using the Lidcombe Program with families who speak languages other than English, so I am confident that this approach will be successful.