modal verbs in reported-speech backshift
Solution 1:
Maybe a native speaker could and maybe they couldn't, depending on what pack of rules you've been saddled with. The list above isn't really about modals; it's about the necessities of events and how one tries to describe them in words. Modals are very helpful in that, for sure. Also, it doesn't matter a whit whether she later discovers her mistake or not. She said what she said, and it must be reported correctly; syntax is not a Bayesian enterprise.
MUST
In direct speech, That must be wrong is the Epistemic sense of must, meaning it's logically impossible (rather than under an obligation to be wrong, which is the Deontic sense must.
In indirect speech, both (1) and (2) are epistemic senses (we tend not to use deontic must for simple courtesy), and also acceptable, though (1) seems unnecessarily complex, and the perfect infinitive changes the point of view to the present, instead of the past time, when she said it. In an indirect quotation, one must be careful not to report anything that implies, suggests, presupposes, entails, or otherwise includes something that was not in the original.
MAY
All of these are epistemic, of course; it could hardly be otherwise with an abstract subject and predicate. As above, (1) is overthink but grammatical, and means the same as (2), also acceptable. (3), however, is different. If it is something that changes correctness, then (3) does not report what she said -- instead, it implies that she said it was still correct, which is different. So I'd say it was unacceptable.
The sentences you assign to
WOULD
are very different from the ones with the predicate (be) wrong in the first two groups. Naturally, sentences with different predicates combine differently with modals. These sentences are complex, not simple, in direct speech already. In reported speech, there are so many ways available that some of them have to be monitored.
Here none of the choices are correct, because the constituents aren't combined. A correct reported speech version is
- She said she would help you but couldn't.
The conjoined clauses she would help you and she couldn't help you are joined after said by Conjunction Reduction, which keeps them that way. By contrast, (2), the closest to acceptable of all the choices, is
- She said she would help me, but she couldn't.
The comma and the lack of full conjunction reduction leads to a suggestion that the "she said" part extends only to the she would help you as a promise, and then an unreported part meaning but she actually couldn't, which is not what she said and not what you'd want.
There's more; there's always more with Modals. But I desist.