Definitely not, IMHO. Nothing against the Vereina tunnel in general, though; but it is built mainly for cars and mobile homes.
I just don't see a point in riding 'The Alps' by using tunnels if there are passes around (and you don't have to because of early / late snowfall).
I would suggest to take the route over Flueela pass instead. Will take you a couple of minutes longer than using the railways through the tunnel (waiting for the next train not taken into account), but worth it even if the weather should be less than favourable.
Or, if you fancy narrow and bumpy scenic roads, ride almost all the way down from Davos to Tiefencastel, turn left and take the road to Albula pass. Then left again down the Engadin valley. This would be my favourite if time is not a big issue.
Nauders - Innsbruck is just boring BTW. *Lots* of traffic, speed traps, and seldom a chance to overtake. Take in as many scenic routes as you possibly can before that stretch of road ...
Have fun,
Stefan