Hi Anna-Marie and welcome to the Forum
I've walked the Le Puy route twice and it's beatiful but tough. I walked during April/May and that was a busy time in 2005. I heard somewhere that the GR65 wasn't too busy in August so starting mid to late August shouldn't be too bad for accommodation. Regarding heat, while it may have built up through the summer, remember that Le Puy is quite high and you will be walking across the Auvergne plateau at about 1000m for about a week before dropping into the Lot valley. By the way the temperatures can plummet at night on the plateau even in August, as I remember from camping there many years ago.
Regarding September, I wouldn't expect it to be too cold, in fact I would expect it to be quite pleasant walking weather, but you could access weather records for both August and September for the areas you will pass through. Accommodation - the GR65 is popular and well stocked with refugios, gites d'etape, etc so I wouldn't expect that there would be a problem with closures in September. Also, since many people walk in the Spring and others (French and most of Europe) have just had their summer holiday, I think accommodation would be fairly easy to get, ie not crowded. This might also be the reason why I heard that August is not too bad - people on family hols rather than long hikes.
Good luck with your planning and feel free to send more questions.
Buen Camino,
Brendan