tell us more about your background. There's the backpacking part and the mountaineering part. Consider the Pyrenees are rugged mountains and the GR11 is close to mountaineering sometimes. It goes over really steep, rugged terrain in many spots. Nothing technically difficult but if your background is in backpacking in good trails it's better to be aware of this. I'm not sure how much of this is reflected on the literature.
As far as backpacking, the GR11 is not difficult. It takes around a month+ to complete so it's not really what Francis Tapon is talking about in his article which refers more to multi-month trips. Still, what he says is important and deserves attention, it also applies to multi-week trips if you're not used to them. Back to the GR11 from the backpacking perspective, I'd say it's on the easy side: civilization is never too far, supplies are easy to find and the mountain huts (stuffed or not) are usually at hand even in the most "remote" areas. Nothing is really that remote along the GR11.
As an example, I can tell you in Spain backpacking is not too popular, mountaineering is far more common as the default outdoor activity. Most locals who attempt the GR11 are mountaineers. They usually do fine even though they usually take ridiculous amounts of ultra-heavy gear :) as a consequence of their background. On the other hand, backpackers not used to mountaineering may find it difficult, particularly if they expect a well-groomed, easy graded trail which the GR11 is not.
You'll be high in the mountains often, over exposed terrain. In good weather, it's fine but the potential for bad weather is always there and it can get very serious in the high areas.
I'd encourage you to go for it if that's what you really want to do. I'd say you'll be fine if you don't do too stupid things but beware it's easier than it seems to do stupid things, we all do eventually. Be humble in the mountains.
As for season, the easiest is July to September. In June there may still be snow in the passes, it depends on the season. You might still find some in July but that's not as common and it shouldn't be much (but beware! the snow may be easy or very dangerous, it depends a lot on conditions at the very time you go through). August should be snow free but evening thunderstorms are at their worst. September is colder and days are shorter but the weather is usually still stable and there's far less people. July and August get crowded.