If you interpret the bit mask as encoding a finite set $S = \{2^{b_i}\}$ of powers of $2$, you are precisely asking whether there exists a subset of $S$ which sums to $A$ modulo $p$. This is known as the modular subset-sum problem, and apparently there are efficient algorithms for it, for example https://arxiv.org/pdf/2008.10577.pdf

(EDIT: The modular subset-sum problem is phrased as an existence statement, whereas you want to find an explicit subset. At least for the dynamic programming algorithm sketched in the beginning of the linked paper, which already gets you $O(dp)$ time where $d$ is the digit sum of your mask, this should not be a problem, since instead you can tag every element of the intermediate sum set $S_i$ by one choice of subset (or $B$) that gets you there. I haven't checked whether the other algorithms are similarly constructive)

If you interpret the bit mask as encoding a finite set $S = \{2^{b_i}\}$ of powers of $2$, you are precisely asking whether there exists a subset of $S$ which sums to $A$ modulo $p$. This is known as the modular subset-sum problem, for algorithms, see for example https://arxiv.org/pdf/2008.10577.pdf

Since there are primes such that all elements of $\mathbb{Z}/p$ can be written as powers of $2$, this problem also does not seem to be easier than modular subset-sum. If I read the introduction to the linked paper correctly, there are conjectural lower bounds on the runtime that look like $p^{1-\varepsilon}$, so it doesn't seem to get better than polynomial in $p$.

(EDIT: The modular subset-sum problem is phrased as an existence statement, whereas you want to find an explicit subset. At least for the dynamic programming algorithm sketched in the beginning of the linked paper, which gets you $O(dp)$ time where $d$ is the digit sum of your mask, this should not be a problem, since instead you can tag every element of the intermediate sum set $S_i$ by one choice of subset (or $B$) that gets you there. I haven't checked whether the other algorithms are similarly constructive)

If you interpret the bit mask as encoding a finite set $S = \{2^{b_i}\}$ of powers of $2$, you are precisely asking whether there exists a subset of $S$ which sums to $A$ modulo $p$. This is known as the modular subset-sum problem, and apparently there are efficient algorithms for it, for example https://arxiv.org/pdf/2008.10577.pdf

If you interpret the bit mask as encoding a finite set $S = \{2^{b_i}\}$ of powers of $2$, you are precisely asking whether there exists a subset of $S$ which sums to $A$ modulo $p$. This is known as the modular subset-sum problem, and apparently there are efficient algorithms for it, for example https://arxiv.org/pdf/2008.10577.pdf

(EDIT: The modular subset-sum problem is phrased as an existence statement, whereas you want to find an explicit subset. At least for the dynamic programming algorithm sketched in the beginning of the linked paper, which already gets you $O(dp)$ time where $d$ is the digit sum of your mask, this should not be a problem, since instead you can tag every element of the intermediate sum set $S_i$ by one choice of subset (or $B$) that gets you there. I haven't checked whether the other algorithms are similarly constructive)