Welcome!
I have a stock answer I've posted before, but I can copy and paste faster than you'd search it up:
(Note---I have lots of bikes--road cyclocross, hardtail, full suspension. My wife and I have done 5,000 km of varios Caminos, all on hardtail MTB's. You'll find other opinions, but we have not been tempted to bring different bikes.
The route, bike and tires
I looked all over the web trying to figure out if the Camino was best on road or mountain bikes and what tires to use, and never found a decent answer. There are tour groups and road cycling groups riding the route and the paralleling roads who try to make a case for road bikes but I think it would be a big mistake to use anything but a mountain bike with full size mountain bike tires. The route is not difficult mountain biking, more often it is on a dirt road rather than a hiking trail. There is not much pavement, maybe 10% in the first half and 30% for the rest of it. It is a wonderful ride and it would be a shame to miss it by riding the roads. There are people who ride road bikes on paved alternative routes, but unlike France, where the roads are wonderful, much of the route is crowded with high speed truck traffic. We followed the walker's route 95% of the way, and only walked our bikes about 2 km in the entire trip.
I would use a 1.9-2.2 tire with small to medium knobs in the rear and a 1.9 mountain bike racing tire with side knobs and a smoother center in the front. The first year we had a 1.5 rear tire with road tread in the rear and a side knob front tire and bought full sized rear tires halfway through the trip. Last year we had mountain bike racing tires with very little tread and were happy until it rained, when we again bought some knobby rear tires.
Most walkers are glad to share the path with a bike, just ding a bell to let them know you are on the trail and pass them politely.