There are some pleasant rideable sections of the GR10 in the Basque country but once you're into Béarn the paths aren't really MTB friendly. The GR10 through Gourette is less than 5% rideable on the climbs and suicidal downhill. Further east it improves again but not much. I did about 30km of the section through Bigorre on an adventure race and would have happily swapped my bike for a rucksac of the same weight. Even further east things improve (a bit) but it only gets properly rideable about 30km from the Med coast. There is a proper MTB route with guide books on the Spanish side but I've never done it.

If you choose to walk I suggest the HRP which is much more "out there man".

Edit: opps, found this a bit late!