OP, you're not really on the right track looking for GR routes to find bike trails. The GR's are long-distance paths (note that's paths, NOT footpaths!) of one type or another (local, regional, national, international…). Some of them make for great riding, some don't.
Looking for trails on a French IGN map is much the same as on an OS map at home. Off road trails are indicated by black lines. Officially, dashed lines mean "viabilité incertaine" (i.e. could be impassible) but in general I've found that solid black lines mean forest roads/landy tracks and dashed lines mean singletrack. If the lines (dashed or otherwise) are highlighted in red, this means that they are waymarked in some way and is generally a good sign that the trail will actually be there on the ground!
You can take from that advice that the first trails you should be checking out are the dashed lines with red highlighting…
You can also find a lot of info about marked bike trails on the FFC site: http://www.ffc.fr
Note that, as with most civilised nations, there is no footpath/bridleway type distinction in France. French land access is pretty similar to how Scottish access was a few years back (before the Land Reform Act). Basically you can go anywhere you like as long as you act responsibly and no-one tells you not to. Private land owners have the right to ban you from their land for no particular reason and the local Mayor can also pass a "arreté municipal" which may ban particular user-groups from a given trail (bikes/motocrossers/everyone!) but there has to be a sign at the trailhead.