In the 1933 original Polish version of his monograph Tarski says that his method for constructing truth predicates cannot be applied to the construction of a truth predicate for LGTC. The problem is that in 1933 Tarski adopts as the mathematical apparatus of his metalanguages the simple theory of finite types, or equivalently, LGTC.  In this language, supplemented (and even unsupplemented) by a theory of syntax for LCC, Tarski has everything he needs to give his definition of satisfaction for LCC. In particular, the relation of satisfaction for LCC is a relation that can be found in the hierarchy of finite types. (Full set theory is not needed.) And the form in which Tarski defines it quantifies only over sequences of classes and over relations among sequences of classes and formulae (as we may check above), and these are objects quantifiable over in finite type theory.