I don't know how I would feel about engaging the public school teacher for private lessons - but then again, she does know your child already, and hiring teachers as private tutors is common, right? It might be fine.
Buying a piano is a big commitment. We just bought a high-quality digital piano on the recommendation of my daughter's piano teacher but my child is very young and we are committed to at least a few years of lessons. If your son is more about "trying it out" then it's harder to make that financial commitment.
Do you live in a very rural area? I'd be surprised that anywhere decently populated would not have a violin teacher available to you. That said, I also found it hard to find a teacher even though we live in a reasonably sized town. It seems that many private music teachers have not found the need for professional websites. Word of mouth seems to be the way many teachers find students, at least where I live. It makes finding a teacher pretty tricky. If your son is more interested in the violin, I'd really try for that as he'll be more invested in learning.
