Would it be correct to use "kérem" instead of "kérek"?
I think that would actually match "I do not want it" better than kérek does. I would say the sentence with kérek probably better translates as "...but I don't want any." (Better let a native speaker weigh in to be certain, though.)
You are absolutely correct.
With the "kérek" version, you could also add "belőle" - from it. "Nem kérek belőle."