I have a Brompton and have used it for touring. I’ve also cycled the Camino Frances
from Pamplona to SDC, though on a mountain bike, so I can probably speak to both.
It’s feasible to do the trip on a Brompton, and I think you can find trip reports that others have done. There are guidebooks for finding parallel roads rather than riding on the dirt paths with the hikers. That was my preference, even with a mountain bike.
I averaged 2 to 3 walker stages daily, about 50-75 km. That means you’ll largely be alone since you’ll leave walkers behind daily, and there aren’t large clusters of cyclists. Still, if you’re outgoing, you’ll meet others, and it might be fun to slow down, explore the countryside, and do some stages at walker distances. I rode with a fair number of local Spaniard cycle groups and found them to be friendly, if not jovial companions for a few hours.