One month should be enough to ride the Pyrenees and Northern Spain. Start in the East and head West. I wouldn't advise the route of the Raid Pyrenee which is a bit of a dash, there are better routes that take in local loops and scenic but not spectacular passes as well as the big ones. Northern slopes are greener and cooler than south-facing ones.
In Spain you can use the Camino but take a trip to the Picos de Europas, a very pretty little mountain range. The coast is also very good with little fishing villages.
July is peak tourist season so you may find some of the local hotels and pensiones booked up. The Tour de France will suck up every hotel room for miles around. If you decide to camp, you can pick up a tent and some bags locally at one of the discount sports stores such as Decathlon. There is no need to cook if travelling light is your priority.
A note of warning when camping in Pyrenean valleys. Flash-flooding is a serious matter and campsites have warnings and procedures. Note them down and be prepared.
This is classic cycling country with excellent bike shops in every town and a strong local cycle racing and touring scene. Sounds like a great idea.
There are some off-bike attractions such as cave paintings in la grotte de Niaux, well worth the visit.