You may be attacking a straw man. Brown says Oral Law is not mentioned in response to a Jewish argument that requirement to believe in Messiah is not mentioned. His point is that a "non-mention" in Tanach is not necessarily proof that the argument is false.

Brown does not claim there is a verse that says atonement must have blood (other missionaries do). He does claim that every example Jews give of bloodless atonement really has blood if you analyze the case more deeply. This at times forces him, if I recall correctly, to borrow from Oral Torah- e.g. when Nineveh was forgiven by G-d, Brown claims the Sukkos offerings help in this case. I do not recall if Brown had any BIBLICAL argument to prove that Sukkos offerings help.

Even if Brown uses a single standard, the problem is he's wrong.