The Pyrenees are indeed a notch wilder, especially the central part. But September will be fairly quiet in the Alps. If you’re planning to hike during the second part of September, certainly consider the circular ‘Tour du Mont Blanc’: Chamonix is easily accessible from Paris (a plus if you're not proficient in French and Murphy's law strikes) – check http://en.voyages-sncf.com/en/ for train times and online ticket purchases (train tickets are normally released three months before departure).
The route has some superb balcony walks. It would be a stress free, enyoyable one week hike during the second half of September, but do check opening times of the huts – most close down during the second half of September. A good website: http://www.autourdumontblanc.com/en/index.cfm/tour-mont-blanc-accomodations.html
Keep in mind that the Pyrenees has no wilderness status like in North America. But some sections certainly feel wilders than e.g. the SEKI national parks. During season, hut based walking is possible. A lot of huts in the Pyrenees go into winter modus during the second half of September, so double check before setting out without a tent.
I would advice bringing a lightweight tent – you’ll encounter many first class camping sites. A lot of huts in the Pyrenees allow pitching tents close by. In short: a tent will give you more flexibility.
Be prepared snow occasional snowfall during September, both in the Pyrenees and the Alps.
The GR10 is nice, but there are better options. Chance of clouds are also higher on the French side of the Pyrenees (the Spanish side statistically sees more sun, confirmed through consistent anecdotal evidence, including my own).
I would rather focus on the central part of the ‘Haute Route Pyrenees’ (HRP), an unofficial route seeking out the highest parts of the Pyrenees. BPL-member Amy Lauterbach has written an excellent trip report on this forum. Plenty of gps logs on Wikiloc.com
Accessing trailheads in the Pyrenees is a bit more complex than accessing Chamonix. Some possible exit/entry points in the central Pyrenees (entry/exit from the French side):
– Col de Somport (SNCF bus from to Oloron-Sainte-Marie train station)
– Cauterets (SNCF bus from/to Lourdes train station)
– Bagnères de Luchon (SNCF bus or SNCF train from/to Montréjeau train station)
– Arinsal, Andorra (Andorrabybus.com from/to Toulouse train station or Toulouse Airport)
The train stations mentioned are on or close by the main Toulouse-Bordeaux line.
If you’re interested in a circular route in the Pyrenees, the ‘Carros de Foc’, on the Spanish side is apparently a superb trail. Camping is however prohibited on large sections as it runs through a national park with camping restrictions). I have no experience with this route.
RE guidebooks: Cicerone.co.uk is IMHO the best publisher of English guidebooks regarding both the Pyrenees and the Alps. Many of their publications can be purchased as Adobe ebooks (read: immediate access and weight saving on the trail).