This question certainly makes sense to ask, butand the answer is no, because of how flexible the definition of "logical system" is.
First, your definition of B1 < B2 can be simplified: "T(B1) is a subset of T(B2)" is equivalent to "B2 proves B1".
Now, suppose we form a logical system whose only statement symbols are xn, where n can be an integer, and (countably many) rules of inference stating that xm implies xn when m > n. If A is the set of axioms {x1,x2,x3...}, then there is no minimal set of axioms implying A (and P(A)). This is because a set of axioms B will imply A if and only if it contains a sequence of axioms with subscripts approaching infinity, in which case we can always remove finitely many axioms from B so that it still implies A.
(FYI, systems with infinitely many axioms and/or rules of inference are typical of modern mathematics; ZFC, the most commonly accepted foundation, is itself not finitely axiomatizable.)