I believe that the の here is replaceable with が here
No, it isn't. Just because の appears in a relative clause does not necessarily mean it's a subject marker. You cannot replace this の to が. 歯が治療をする医者 is still "grammatical", but it means something funny like "a physician whose teeth treat patients"!
This の in 歯の治療をする医者 is simply a noun-linking particle ...
Your analysis is correct.
The basic sentence here is:
Oh, this gown is also cheap, right?
This is a perfectly formed sentence in its own right.
But you want to be able to describe the gown you're talking about. The thing that does the describing is the relative clause (look this up): 水色の縁がついた. This is also a ...