Given $k\in\Bbb N$ with $k<(\log_2N)^{\frac1\alpha}$ where $\alpha>2$ is fixed and $N$ being some integer such that $$N<\prod_{i=1}^k\pi_i^{a_i}$$ where $\pi_1,\pi_2,\dots,\pi_{k-1},\pi_k$ are unequal primes less than $\log_2N$ with some natural numbers $a_i$ and suppose we know the smallest non-negative member in each of the residue classes $$N\bmod\pi_i^{a_i}$$ at every $i\in\{1,\dots,k\}$, then given an integer $M<\log_2N$, is it possible to compute smallest non-negative member in the residue class $$N\bmod M$$ without first reconstructing $N$ by $\mathsf{CRT}$ or any other means?

If so is there an efficient algorithm (efficient here is running in $O( (\log_2N)^{\frac1\alpha}\log^cM)$ arithmetic operations where $c\geq0$ is fixed)? By arithmetic operations, I mean $\{+,-,\times,\bmod\}$ operations without regard to size of integer