I wouldn't suggest calling it the french cafe, unless you make it a joke, and give all the non-french things you serve french names, in the way that they've done in france (despite the language police and french language academy that wants to keep foreign words out of france) where they put the french article in front of the english word (le pullover, le cake, le plumcake)
If you want to pull it off really well you'd have to find out how they do call things like muffins and pies in france (a "tarte" is not really a pie).
For instance is it "Le muffin" or "la muffin", (muffins are pretty universal in europe now, though they often seem more like cupcakes, but it might be cute for people to see "le muffin" "le muffin de mais" (corn muffins). But you'd have to be careful not to be misleading (tarte aux pommes is a flat tart with apples laid on top and apricot jam spread on it) My french is nowhere good enough but hyou would have to call it something like "tarte aux pommes a l'americaine" or maybe "le pie des pommes" (don't use this, the grammar is probably wrong, but you get the idea) -
It would be tacky to have a poor french translation with the wrong gender or case or usage. Just to give you an example of how bad it can sound, there's a butcher specializing in pork products here in rome and you see the trucks around occasionally and they call it "Pig's house" - which was clearly some italian trying to translate "la casa del maiale" which sounds perfectly fine in italian, but he didn't consult a mother tongue speaker of english. In english you should say "the house of pork" . There are hundreds of examples of this around and they always look bad. Use french or any language you like but make SURE you're using it correctly!