I would say that you should first try to do this without building out a .fr domain. In your position, and with some experience of this already under my belt, I would look to build pages relevant to France in a /fr folder, and use Webmaster Tools to inform Google that this is where the geographical target for that folder is.
There are many other things to think about, just a few of them are;
1. Filtering of results by the user will have an impact i.e. the search for sites in French/ search for sites in France choice that users will asked to make. It's great that the site shows up in your tests, but maybe it would disappear to some users if it's not hosted in France?
2. You could still buy an appropriate .fr domain, build out a little relevant content hosted in France, link to this from your obviously reasonably authoritative main site and see if you can start to get the new domain to rank and "fill in the blanks".
3. You don't say if you translated the key phrase you used, and I'm assuming translation would be relevant. If it is, and excuse me if this is obvious, you'll need to do all of your keyword research again. Just translating from say, English probably won't cut the mustard.
I seem to recall there was a Whiteboard Friday on this topic quite recently, you should look that up.
