This is the “on-this-page” search option for safari, and I think it would work well for in-note search on bear. Thoughts on where to access is same with the side bar (but without the search icon), or if icon is needed be, maybe have it on the actions sheet. Keep scrolling at the top of the note to invoke search, just like it does on the side bar, and then search comes up on top of the keyboard to start searching.
One advancement I wish Bear had was that the two find feature is incorporated into one. Meaning, when I search from the side bar, I can click the note and do note have to press cmd + F again to invoke search next matching on the document. Make the two finder show up at the same time (have go through option) when I search in the side bar. And I hope it translates the same to the iOS version once it is implemented.