Hi Wendy -
Have a wonderful camino with your friends, walking and biking.
I would strongly suggest you do wear a helmet ... the paths are loose and stony and accidents can happen anywhere - roads, country lanes etc.
The reason why I strongly suggest that you do wear a helmet is that I fell off my mountain bike last April while training for a bike camino. My shoulder and head took the brunt of the fall - I broke my collarbone and scapula and the collarbone had to be pinned and plated. The doctor in A&E told me if I had not been wearing my helmet the accident would have been so much worse - I would have sustained head injuries and I could have died. At the time, and in the months since, I have thought countless times about how lucky I was to have been wearing my helmet (it's the law here in Australia) - wearing my helmet prevented me from a much worse injury and indeed could have saved my life.
Wearing a helmet is a drag sometimes - your head gets sweaty and you get the worst sort of hat hair ever, BUT, after my experience, I would never hop on a bike without one. You'll be having a shower and can wash your hair as soon as you arrive at the albergue/hostel/casa rural/hotel anyway, so please, play it safe and wear a helmet.
A note on bike caminos generally - have your pack transported each day - as you plan to walk and bike this really is your only option.
Buen Bike and Walking Camino Wendy and best of luck with your preparations -
Take joy in every step and every pedal -
Cheers - Jenny