I was thinking that dropping Moeen is a terrible plan, but then you have Root and Malan who can do some bowling so it isn't the worst idea. Plus if Livingstone plays he can bowl too. Although it is a difficult question as to who you go with Ali vs Livingstone if it is for the same spot. Eng would pick Moeen.
There won't be too much of an issue about pace in this series - pitches in NZ tend to favour bowlers who hit the seam consistently at around 80mph just fine, and we are okay in that department.
But if we don't have Stokes, we cannot rely on Ali as one of a four man attack, so we either drop a batsman to play five or drop Ali. Based on his performance in the Ashes, I have no issue with him being dropped - indeed, I think its probably worth looking at other options, even if he will probabaly return to the side for the start of the UK summer.
So, that leaves either:
Cook, Stoneman, Vince, Root, Malan, Stokes, Bairstow, Woakes, Overton, Broad, Anderson
Or, a better balanced side:
Cook, Stoneman, Malan, Root, Stokes, Bairstow, Livingstone, Woakes, Overton, Broad, Anderson