A dilemma for the deflationary theory of truth

Is the claim that truth is completely characterized by the deflationary theory itself a part of the deflationary theory of truth?

If yes, then the deflationary theory of truth has a problematic kind of self-referentiality: it contains the statement:

1. The theory which includes (1) and statements X-Z completely characterizes truth.

But it is not clear that the content of (1) is sufficiently determinate. It seems that (1) suffers from a similar semantic underdetermination to that suffered by the truthteller sentence:

2. Statement (2) is true.

Suppose now that the claim that truth is completely characterized by the deflationary theory of truth is not a part of the deflationary theory of truth. There are now two options. Either the completeness claim is or is not known to be a consequence of the deflationary theory of truth. If it is not a known consequence of the deflationary theory of truth, then we have a problem for deflationary theorists who assert that the deflationary theory of truth is complete. For if they do not know their assertion to be a consequence of the deflationary theory of truth. But if it is not a consequence of it, then the theory is not complete. So it seems unlikely that they know the theory to be complete, and they are asserting something something they do not know.

Suppose that the completeness claim is known to be a consequence of the deflationary theory of truth. But the deflationary theory of truth without the completeness claim consists merely in a claim as to what the truth-bearers are and all the instances of the T-schema. But the completeness claim simply does not follow from these. For there are multiple predicates, with "is true" being only one of them, that satisfy the deflationary theory of truth: any predicate extensionally equivalent to "is true" satisfies the deflationary theory of truth just as well. And such predicates are myriad. For instance: "is true and is not false", "is believed by God", "is true if 2+2=4", etc.

Perhaps if we add to the deflationary theory that "is true" expresses a very natural property, then we can rule out some of the alternate predicates. But "is believed by God" appears very natural as well. So at least the theist can't take this way out.