'Knows' contextualism derivative from 'believes' contextualism?

Here's Brian Weatherson:

[B]elief ascriptions and knowledge ascriptions raise at least some similar issues. Consider a kind of contextualism about belief ascriptions, which holds that (L) can be truly uttered in some contexts, but not in others, depending on just what aspects of Lois Lane’s psychology are relevant in the conversation:

(L) Lois Lane believes that Clark Kent is vulnerable to kryptonite.

We could imagine a theorist who says that whether (L) can be uttered truly depends on whether it matters to the conversation that Lois Lane might not recognise Clark Kent when he’s wearing his Superman uniform. And, this theorist might continue, this isn’t because ‘Clark Kent’ is a context-sensitive expression; it is rather because ‘believes’ is context-sensitive. Such a theorist will also, presumably, say that whether (K) can be uttered truly is context-sensitive.

(K) Lois Lane knows that Clark Kent is vulnerable to kryptonite.

And so, our theorist is a kind of contextualist about knowledge ascriptions. But they might agree with approximately none of the motivations for contextualism about knowledge ascriptions put forward by Cohen (1988), DeRose (1995) or Lewis (1996). Rather, they are a contextualist about knowledge ascriptions solely because they are contextualist about belief ascriptions like (L). Call the position I’ve just described doxastic contextualism about knowledge ascriptions. It’s a kind of contextualism all right; it says that (K) is context sensitive, and not merely because of the context-sensitivity of any term in the ‘that’-clause. But it explains the contextualism solely in terms of the contextualism of belief ascriptions.

I think that the kind of view Brian describes is very interesting and worthy of more attention than it's gotten. But I'm puzzled by his characterization of it. How is it that the knows-contextualism is explained by the belief-contextualism here? Belief-contextualism is a view about the word 'believes', which does not typically occur in knowledge attributions, the subject of knows-contextualism. So how could belief-contextualism explain knows-contextualism? In what sense is the theorist in question "a contextualist about knowledge ascriptions solely because they are contextualist about belief ascriptions"?

One might motivate knows-contextualism in a way reliant on belief-contextualism. It might look something like this: belief-contextualism is true. Therefore, for some situation, there are two contexts C1 and C2 such that in C1, 'S believes p' expresses a truth for that situation, and in C2, 'S believes p' expresses a falsehold for that situation. In some such C1, 'S knows p' also expresses a truth for that situation. In all contexts, 'S knows p' expresses a truth only if 'S believes p' expresses a truth. Therefore, in C2, 'S knows p' does not express a truth. Therefore knows-contextualism is true.

That argument gets the job done, but it doesn't seem at all like it must be motivating the combination of views Brian discusses. Why should one run through a metalinguistic principle about the relation between 'knows' and 'believes'? It seems to me to be at least as plausible that the corresponding views about 'believes' and 'knows' are motivated in parallel by considerations about propositional attitude ascriptions in general. I agree with Brian that someone who thinks (L)'s content depends on context for these kinds of reasons is also likely to accept the same for (K). But I don't think there's any reason to think the resulting knows-contextualism is in any sense derivative from the belief-contextualism. It's just that the same kinds of data motivate both views.