I've had wonderful birthdays on the Camino, in fact my 54th, 55th, 56th, 57th, 58th, 59th and 60th! Last year I left earlier than usual (mid June) so for the first time in years I had it at home although I usually don't make a big deal, just dinner with family.
Two Caminos birthdays stick out. On my first Camino I was in Sahagún on my birthday and I invited a Catalán woman I had been walking with for a drink. When the bar owner heard that it was my birthday she came out with empindada for us and mine had two candles - a 5 and a 4! I still have them 8 years on.
The third one I will remember forever. It was when I was a hospitalera in Grañón. My fellow hospitaleras, Jesús the priest, Suzanna the baker and all the 70+ pilgrims staying the night made the day and evening so special. Can you imagine happy birthday being sung in about 6 different languages! There were ballons and decorations everywhere!
But putting this all aside, in your case I would say walk in October. The weather should be better (I've walked the Norte and Plata/Sanabrés in October and had lovely weather) and you don't have to worry about albergue closures. Xunta albergues in Galicia are known to be open year round but others may close by November.
