I've never done this route. But for many years, I was an avid cycle tourist. I spent six months riding around Europe when I was in my early twenties, and more short trips than I could count. I've toured on touring bikes (Mercian King of Mercia, Dawes Galaxy) and various old mountain bikes.
It really comes down to the distance you're planning to travel per day, and whether or not you have company. When I've toured with mountain bikes, the daily distances were lower, but I had fewer punctures and roadside repairs (tyres should wear out faster on an MTB, but in practice that's never been an issue because I rode with good tyres. Also, a mountain bike (at least the kind that works best for touring) has the advantage that it fits into the environment and attracts less attention, as the best ones are now quite old.
If your friends want to ride a touring bike, you should have one too. You will have to work harder to cover the same distances, which is no fun.
On reflection, I would stick with the MTB, and keep my daily distances lower. Give yourself time to stop and explore. Don't go clipless, platform pedals and walking shoes or Teva sandals are better for walking around. Buy an old, steel framed bike in good condition, and build strong wheels for it. Oh, and buy a Brooks saddle. And spend the money you save on more travel time! You will thank me later.
