Use the object pronoun or the subject pronoun as the relative pronoun heading a restrictive clause that employs a transitive verb and a linking verb?

Solution 1:

James is the man who we know ____ is who won it.

Here, "who" is not object of "know" but subject of the embedded "is" clause, marked by gap '___' .

"Who won it" is a subordinate interrogative clause functioning as predicative complement of “be”.

We understand "we know James is the answer to the question 'Who won it?'", where "James" is the person the whole NP refers to.

Better to avoid too many who's if you wish to sound more natural. I would simply say

James is the man whom we know to have won it.

Having said that, some say that it is ok to use who instead of whom in spoken language.

You can understand your sentence in this way:

James is the man about whom we know that he (has) won it.

Another possibility to use the subject pronoun is:

James is the man who is known (by us or by all or by anybody) to have won it.

However, as they say the shorter the better, you could probably just say:

James is the man known to have won it.

but only in the case when the fact that we know it is not so essential to the meaning of the sentence, and this depends on your wider context.