Peaceable Projects Inc. is a U.S.-based non-profit group that brings the vast resources of the wide world together with the ongoing needs of the people who live, work, and travel on the Camino de Santiago pilgrim trail network in Spain.
My vote would be Monday. Last month, I found it much easier to get rooms in San Martin Pinario across from Cathedral on 2 different Sundays and Mondays. All pilgrim rooms there were booked many weeks ahead for Fri/Sat. Tourists tend to leave on Sundays, and all hotels/albergues in cities I stayed along CF and North Coast were easier and cheaper to book for Sun/Mon. Guaranteed a lot of businesses will close all over Spain, even Madrid, Saturday at noon and will not open until Monday (unless Monday is a holiday). Everything I booked except Santiago and Madrid, I booked same day, and usually in the afternoon. Most towns I just walked into and got albergue on the spot. Never had any trouble finding a bed in that 37 day adventure, but did find numerous businesses closed weekends and holidays and mornings before 0900. Consider also that bus/train/plane schedules vary from weekends to week days.