Only Who did you know that looks like Bill? is unacceptable because who is subject of the nominal that clause. that must be omitted to prevent it being initially misinterpreted as subject of the following verb.
Compare the following optional that,
Who do you expect (that) they have chosen? (who is object in the nominal that clause)
She told me how she thought (that) the machine worked. (how is adverbial in the nominal that clause)
I had expected to hear from a native speaker of English, but thank you all the same.
It is generally known that we can’t extract something from the subject position in a smaller clause when there is a complementizer (like “that”). So, delete “that” and (3) will be fine.
(3) Who do you think (*that) was chosen?
As for (2b), “who” comes from the object position of “know” but not from the subject position of “looks.” In other words, the “that” clause is a relative clause modifying “who” and not the object noun clause of “know.”
It’s difficult to think of an example sentence where “know” means “recognize,” but if it works in (2a), I’m wondering if (2a) is acceptable or not. The reason: when “know” works as a “bridge verb,” there seems to be a restriction on its meanings. And I’m assuming that English-speaking people will disapprove of (2a). But I’m not sure. That’s why I posted this thread here.