July/August
Hi Jessie,
I think the later in the summer you can walk the better.
We walked in Italy last July and the temperatures reached 40oC a number of times. Coming from a hot climate (Durban, South Africa) none of us found it unbearable but the locals thought we were completely "pazzo" (crazy or Loco) to be walking in the mid-day sun.
Hydrate well, use lots of sunblock and remember, it it hotter at 3pm than it is at mid-day so try to get to your destination before mid-afternoon.
If you only have 3 weeks, perhaps you should consider skipping the Meseta which can be an oven in summer.
Besides the heat, another reason why so many websites - CSK et al - warn against walking in August is because many Europeans take their annual holidays in August and the camino can be rather crowded with pilgrim groups.
Besides all that, huge numbers of pilgrims walk the camino in August and love it - and I'm sure you will too!