Learn English – Must know vs. should know

He must know, then, she thought. Someone at the co-op has told him.

The two marked sentences are both direct "speech", although Oates does not mark them with quotation marks—probably because they are not spoken but only thought. The must here therefore represents present tense.

But the past tense of must is not had to, though had to may substitute for it. The past tense of must is must^†.

In any case, there is no justification for using either may or might here; either would completely change the meaning.

^† Historically, in fact, must was the past tense of the verb mote, which long since disappeared from use. Must appears to be following in its footsteps, being more and more replaced by the more flexible quasi-modal HAVE to.

Context, context, context! Either 1 or 3 could be correct.

1 is employed in a context where your discourse concerns the time when she was living in London and mention her move to New York only to provide a 'timeframe' for her London residence; for instance:

She was living in London for several years before she moved to New York, and found British accents (once she was able to understand them) much more attractive than that of her native Minnesota.

3 is employed in a context where your discourse concerns the time when or after she moved to New York and wish to mention her prior residence in London as the source of some state which obtained at that later time; for instance:

She had lived in London for several years before she moved to New York, and found it difficult to understand even midwestern American accents when she returned.