On one hand, looking at some of the road conditions, an MTB would be nice to have on the really gravelly sections, especially on the descents where the disc brakes will come very handy.
On the other hand, MTBs are great on trails but simply are no fun to ride on the tarmac. A cross bike would be a lot more fun to ride on the vast majority of the trails, as well as on the climbs.
There are going to be a few crappy segments:
1/ Parts of the Manali-Rohtang route, where landslides etc make the road very muddy. Here, I think I can manage on a cross bike with knobbies
2/ The descent from Rohtang - here, a MTB's superior braking will be much better for sure.
3/ Segments of the climb to Baralacha - not an issue, as it is going to be dry and bumpy roads on climbs isnt really a big deal
4/ Descent from Tanglang La - again, an MTB would be preferable but this shouldn't be as big a deal as it is a pretty gradual descent and the roads are quite wide, from what I remember.
So does it make sense to take an MTB or should I just take my cross bike and 700x35 knobbies??? Or even the LHT, which can fit 700x45?