On submultisets of given cardinality and bound that sum to $0$
Assuming that $c$ is fixed, this can be solved using dynamic programming in quasipolynomial time. We compute inductively all possible sums of all submultisets of size $k$ for all $k \leq \lceil m^\alpha \rceil$. Since each such sum is at most $2^{\log^c m} \lceil m^\alpha \rceil$, this algorithm runs in $O(2^{O(\log^c m)} m^{O(1)})$, which is quasipolynomial. This suggests that the problem is not NP-hard.