I took my cross bike for a week of road riding in the alps last year, and it did well. It was set up as more of a road machine with slicks and compact chainset, but it coped well with climbing and descending and long rides. Having discs was very confidence inspiring, especially going down alp d-huez and all the others. No need to worry about overheating brakes or stressing the forks, it’s what they were built for! The pads will wear a bit on long descents, and I had to adjust mine (pretty average BB5’s) to keep them feeling good.
You don’t need super-low gearing, a 50/34 compact and 12-28 cassette is good. The hills aren’t so steep (generally 7-10%), but they are all long, so you need a gear you can spin at for some time, and to bail into when you’re knackered (it will happen!). As above, the road surfaces are way better than the UK so you can run high pressures and narrow tyres easily.