Could a being be defined as such that its transworld identity is identical to its in-world identity?

Question

I was reading through Collier[21], which is about Lewisian theism, alongside the SEP article on transworld identity, and have assumed that:

1. The concept of transworld identity (TI) is not necessarily identical to the concept of in-world identity (II).

I suppose a simple argument for (1) would be, "If TI ≡ II, then there would be no problem of transworld identity at all." The SEP article seems hesitant about the existence of such a problem, noting how three reasons for the problem's existence can be well-undermined but apparently positing that it is a live issue what the criterion of TI is (or: supposing that it is a live option to debate that criterion rather than abandon TI altogether).

Now, maybe there's an infelicity in (1), or a redundancy, as if identity is necessary in some relevant manner. So we might say:

2. ◊(TI = II) and ◊¬(TI = II) but ¬◊(TI ≡ II), so □¬(TI ≡ II)

... where the equality relation is weaker than identity. Perhaps we could say that identity is generalized enough to be necessary, but equality is then otherwise the same relation but over particulars. (I have seen so many variations on equivalence symbols/concepts as to be lost when it comes to many, or even most, of the details.) But then can we axiomatize a being such that:

3. ∃x(TI(x) = II(x))?

... which is to say, is there a being such that its TI is particularly equal to its II, or the distinction between TI and II is insubstantial for this being? We could, perhaps, reframe the question in terms of Lewisian counterparts:

4. Letting "CI" be "counterpart identity" (identifying something a as a counterpart of b), ∃x(CI(x) = II(x))?

Maybe the intended understanding of CI rules this out (I half-assume so, but this assumption is based on my hope that I understand this topic well enough to be so bold as to assert what is impossible for a definition of CI to sustain). We would be claiming that it was possible for some x to be such that x's-having-nontrivial-counterparts is the same as x's-not-having-nontrivial-counterparts, maybe (taking everything to be trivially its own counterpart nevertheless). Against even (3), indirectly, I wonder though if we should have to identify = and ≡ as themselves having transworld and in-world types, which would lead to this implicit possible(?) counterargument (an argument from the metatheory of (3) failing to solve the following questions):

5. ◊(TI =_TI II) or ◊(TI =_II II)?
6. But the metatheory of (3), though it allows us to pose the question of (5), doesn't have an answer for (5).
7. Therefore, the metatheory of (3) generates an internally unsolvable problem, so this metatheory "corrupts its own file" (so to speak) and we should not try to imagine that the property of TI could ever be the same, as a particular (a trope, perhaps), as the property of II.

Perhaps (5) is solvable in this sense, though: for some x such that (TI =_TI II), then for this same x (TI =_II II), and vice versa. However, a subsidiary text on the SEP mentions critiques of Lewis' semantics as carrying a "deviant and unmanageable logic", a criticism that looks easily levelable at the above.