It's the class of every spacetime with the property. Examples besides the Kerr spacetime are the universal covering of anti-de Sitter spacetime, the Reissner-Nodstrom spacetime, even a simple Minkowski spacetime rolled up along the temporal axis (or in fact any spacetime with CTCs).

Thanks for the examples, that's what I suspected, though I find the CTC examples dubious at best, as you appeal to a much stronger impossibility to justify a weaker one. I am not a stickler for global hyperbolicity, I can certainly imagine topological and/or geometric instantons "magically" appearing and disappearing. These don't cause infinite backreaction the way CTCs do.

If you're interested, here's a nice paper showing that a Malament-Hogarth spacetime can be constructed that satisfies various criteria of physical reasonableness (energy conditions, stable causality, etc.).

It does indeed attempts to address most of the issues, but not the divergent emissions one, which seems mutually exclusive with non-divergent red shift. I am even fine with the "requires infinite energy" issue, since I can certainly imagine pumping energy through a whitehole from some other inaccessible spacetime (or some other instanton-like event).

Does rigging boundary conditions in this manner take us outside the realm of physical possibility?

My interest is whether some hypercomputational construct can be embedded into our universe (which is roughly of the expanding FRW-dS type), not whether some other universe where entropy can decrease can perform these tricks. The reason, again, is that if you use much stronger assumptions to justify something weaker, the argument becomes much less interesting. In an extreme case "because DM decided so" would trivially support anything you want.