So what if a woman isn’t necessarily independent, but has no wali? (as is the case with my wife; she lives and depends on her parents, but they’re kafrs)

And expanding on that, I read a hadith in al-Kafi that said the ‘if she’s independent’ thing, but added "So if she so desires, she can appoint a wali" does this mean she MUST or just that it’s an option if she wants, but doesn’t have to?

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One hadith is the following:

وقال ( عليه السلام ) : الإِسلام يعلو ، ولا يُعلى عليه ، والكفّار بمنزلة الموتى ، لا يحجبون ، ولا يرثون.

It basically means Islam supersedes and nothing supersedes it, and kafirs are like the dead, and as an example of that, they do not inherit.

Just think about it: how would Allah put the fate or a Muslimah in the hands of a Kafir? Naturally, every Muslim who comes to propose might be rejected by the kafir father. Allah would never put a believer in such a situation. No scholar in history has doubted this law. There is absolute consensus that a kafir parent has no wilayah on his Muslim daughter.

If a woman doesn’t have a wali, she is her own wali. She doesn’t need anyone else’s approval. If she likes to appoint a wali to represent her that’s fine, but she doesn’t have to.