I don't think you can be a Cabinet member and a member of Congress. As Cabinet members are officers of the United States and you cannot hold another office while also being a member of Congress (Article 1, Section 6).
It's also why the Presidential Succession Act may be unconstitutional. As the Constitution specifies that only an officer may be designated a successor and members of Congress are explicitly excluded from being officers.
Given that the first time the Speaker of the House and the Senate president pro tem were proposed as successors under Article II was during the literal 1st Congress in 1792, I think it’s pretty unlikely that any modern court would find that to be unconstitutional.
“Many of the actual people who signed the Constitution thought this was okay and demonstrated their belief by introducing legislation to that effect” is about as ironclad a precedent as you can hope to find.
Even when the 1792 version of the bill was signed, the issue was brought up. By the "actual people who signed the Constitution".
And we've never had to actually test the Presidential Succession Act, which is where the courts would decide if it actually meets the Constitutional standard. The fact that it is a law, has been a law for a while, and a version of it has been a law since nearly the beginning of the country has no bearing on whether or not that law is legal.
This is true! But it’s also immaterial to the point I’m trying to make.
I’m not arguing for or against the de jure constitutionality of the law; I’m saying that it’s unlikely any modern-day court would find it within themselves to overrule the de facto state of affairs that was created by the framers themselves.