(Not a teacher)
I'm not at all as knowledgable about these matters as many people on this site, however, something occurs to me:
I would have though 'we' is the subject and 'last night' is the object of 'saw. 'Which' is just a subordinating conjunction. You can remove it, yes, but the subject remains the same; who did the seeing? we did. 'Which' here stands for the noun phrase in the main clause - 'The movie'.'The movie which we saw last night wasn't very good.'
In this example, 'which' is now the object, so it can be left out of the sentence.
As you said, relative pronouns can be subjects of a relative clause:
He was the boy who kissed the girl.
In such cases, removing the relative pronoun is possible, but more controversial: 'He was the boy kissed the girl'.
I can't think of any examples where a relative pronoun is the object of a verb in a relative clause.
Anyway, best to wait and hear what those who know more say...