I’m in Pau which is 110km from your route at the closest point: Cap Breton. There are lots of MTB trails around Pau but the cycle paths are few and far between. The coast to coast route which goes through Pau is mainly roads with an insignificant amount of proper path added each year.
The Vélodyssée route down the west coast is the best route to the Spanish border from where you are. It’s a bit contrived in places but generally safe. The surface is usually OK except for some rooty sections in the pine forests. I sometimes found myself doing a lot of east-west detours when I wanted to head south, the alternatives were more dangerous though so I mainly stuck with it.
From the Spanish border you’ll find yourself on roads a lot of the time. The North Coast Compostelle (St James way) touring bike route is probably your best bet for your section to Bilbao on a touring bike with trailer. The MTB and walking routes are quite physical with steps, stiles and very steep climbs.
Most of the tourists will be heading home on 27/8 so you’l be riding through places with a nice end of season holiday feel but no trouble finding places on campsites. I avoid wild camping in the forests when it’s dry but treat the rest halts along cycle paths as free campsites.