The first half to three weeks of September are very busy. From about the third week in September, departures from St. Jean Pied de Port start to drop off.
Days are still hot and dry. Or, they can be very wet. It depends.
The nights start to get cool. Not wintry, just nippy. You will need a fleece for the evenings, as well a good pack poncho - that covers your rucksack too.
Check with the SJPdP Pilgrim Office for the stats. MonaSP (here in the forum) is a manager there. I know they keep good stats on departures, or at least of those pilgrims who register before starting out.
You could ask her in this thread - OR - send a PM - Private Conversation. Mona is always very helpful and responsive.
IMHO, I would plan on departing SJPdP in the last 10 days of September. Unless you are up to walking 27 km each day on average, 30 days is not enough to walk the full 800 km distance from SJPdP to Santiago.
There are three ways to do this:
1. Find more time to do the entire route;
2. Start at Pamplona, Logroño, etc., instead; or,
3. Do a “leap-frog Camino.” This is where you walk, then bus to leap ahead a few days, then walk again. Repeat to suit your schedule.
Doing a “leap-frog Camino,” you can easily pick up a week of time this way - but you will miss a lot of the local flavor of the Camino. Also, if you started with a “Camino family” in France, your more rapid rate of movement across Northern Spain will certainly leave them behind.
Personally, I would opt for more time and a conventional pace. I suggest 35 days as a good trade off between haste and taking the time to be able to experience the full
Camino Frances.
Hope this helps.
Tom