Solution sketch:

Let real coins weigh $g$, and fake coins weigh $g+x$.

Since we put an equal number of coins on each side at each weighing, the results of the $m$ weighings will be $a_1 x, a_2 x, ... a_m x$ for some integers $a_i$. We don't know $x$ in this case, but we can scale all the results so that they are relatively prime integers. Each unique result then can be mapped to an original bag, thus the problem reduces to counting the number of valid tuples relatively prime tuples.

We can map a tuple $(a_1, a_2, \ldots, a_m)$, with $gcd(a_1,\ldots,a_m) = 1$ onto each bag. In the i-th weighing, we will put $|a_i|$ coins on the left side if $a_i < 0$, otherwise, we will put $a_i$ coins on the right side.

To count this, we can use the mobius inversion formula to do inclusion/exclusion.

Time complexity is $O(k (log k + log m))$.