There are alternative set theories that allow for a universal set, e.g. NF(U), positive set theory and and topological set theory. There are also alternative set theories like ZFA that allow for the Axiom of Foundation to fail in a strong way, e.g. by satisfying Aczel's anti-foundation axiom. I am interested in alternative set theories that combine both of these properties.

In particular, I am looking for a set theory in which the following equation has a solution:

$S = B \times \mathcal{P}(S)^A$

Here, $A$ is a countable (possibly even finite) set, B is an infinite set, and $\mathcal{P}(S)$ denotes the power set of $S$. I believe that the simplified formula $S = B \times S^A$ can be solved in ZFA, but the addition of the $\mathcal{P}$ to the equation makes it unsolvable in ZFA due to size considerations. However, I suspect that an alternative set theory with a universal set might overcome this problem.

Note that the existence of a universal set is not a strict requirement for me. I just suspect that set theories that have it are more likely to be able to solve the above equation.

Can someone point to an alternative set theory that allows for a solution to the above equation?