The White House can’t redact it. The law states that the White House has to sign off on classified info before it’s released. That means sign off on what gets sent.
May or may not be technically true however he can certainly hold it from release until the whole house votes on it until he does get his change requests or redaction in. Be careful on what you assume. There are many derived rights that laws inherit rightly so or plain leveraged.
So it's all or nothing? By way of hypothetical example, the WH couldn't object to disclosing say one paragraph in a 30 page dociument that HSPCI decides to disclose to the public, but has to object to disclosing the whole thing. Wow. I didn't know that. My read of the House rule is that the House is the entity that is effectively declassifying, but the president has a form of veto (that can be over-ridden) in that he can object to the public disclosure.