I walked starting Sept 9 in SJPDP for 5 and a half weeks last year. The weather was great in the most part, hot and dry with a few thunderstorms in the late afternoons in the first part, then getting cooler into the Bierzo and Galicia by mid October. You can get frosts at night in Sept / Oct so you need to have thin layers you can peel off one by one, as it will be cold when you leave in the early morning sometimes. Take care when stopping for a rest or lunch break at altitude as you can chill very quickly especially if you are a bit sweaty - put a fleece on when you stop, even if you feel hot. The heat on the meseta was pretty adverse at times, you
MUST wear a wide brimmed hat, sunscreen and pace yourself. One armed sun tan is a pilgrim hazard (as you are always facing west!) so long sleeves can be helpful. As for the rain, its important to have a waterproof pack liner, so your sleeping bag does not get wet and you always have something dry to change into. Overall I would say that Sept / Oct is a brilliant time to go as far as weather is concerned. Prepare well, travel light and you'll have a great time.
PS The pic of the pool brought back happy memories, I spent my birthday at this refugio, run by such lovely people. The pool was freezing though!!!