If you want to assign probability 1/2 in step 5, you're implicitly doing it by using some symmetry of the problem (namely the symmetry that exchanges you with the twin sitting across from you). But the mathematical issue above means there's no reason to expect that this is actually a symmetry of the problem. If you agree that the number 5/6 doesn't come from looking at symmetries involving twins in step 2, there's no reason to get a second number by looking at symmetries involving twins in step 5.

Are you saying that if I learn that the multiverse hypothesis is correct, it stops being 1/6 and becomes ill-defined? Why would it?

No, I'm saying that it stays 1/6.

The challenge is whether learning this changes your credence, and if so why and to what.

Nothing changes. And a real Bayesian shouldn't believe the angel in the first place.