AJAX with either Flask or Bottle is a great solution for the kind of thing you're trying to do. TBH which you'd choose is very much a case of personal preference, and I wouldn't recommend one over the other. I will say that we use Flask for most of our internal microservices and we're very happy with it -- but I'm sure we'd also be equally happy with Bottle. Perhaps I should recommend Flask just because if you're building your app on PythonAnywhere, we'd be able to give better tech support because we're more familiar with it :-)
Regarding examples -- we're big fans of the "Real Python" tutorials, and they have one for Flask and React that looks pretty solid.