Suppose there is a simple structure $T$ such that it consists of a set of elements $E$ that each have a one-to-one correspondence with a unique element in $S$. Suppose there is a bijective function $F$ that maps from $T$ to $S$. Then $S$ and $T$ equinumerous and contain the same amount of elements.
By definition, a set $A$ cannot be mapped onto another set $B$ with a bijective function unless the cardinality of B is equal or greater than that of A. Therefore, the cardinality of B, assuming bijection with A, is the least possible if cardinality of B is equal to that of A.
Therefore, we are looking for a function to construct the set of $T$, which consists of the same amount of elements of $S$, as well as the bijective function $F$ that connects $T$ to $S$.
Therefore, we are looking for a set constructor, computable function $C(S)$ that will return $(T, F)$.
I will propose that we should look for a μ-recursive function that takes a set and can compute the number of elements in that set. It will construct a set $T$ such that it will induct elements of the natural numbers into $T$ until the elements of the input set are exhausted, and it will also construct a closed recursive function $R$ that can iteratively search linearly over the set of $T$, adding a new level of recursion for each iteration of the constructor function $C$, and correctly map an element of $T$ to its corresponding element $S$. At the end of construction, $R$ will be equivalent to $F$.
If we know that $S$ is finite, then even if $C$ is not closed recursive, if its termination requirement depends on the number of elements on its input set and the set is of finite size, then $C$ should terminate and will be computable.
Therefore, there does exist an algorithm that can construct $(T, F)$ for a finite set $S$ of size $2^n$ such that $T$ is the same size of $S$ (the minimum) and has a bijective function $F$ to map $T$ to $S$.