Should a Cabinet Member meet the requirements of being POTUS if there's no one left?

(Sorry for the tangent, but to understand the question has to have quite a bit of background.)

I recently have seen Designated Survivor the t.v. show. It is actually pretty accurate to a rule that has been posted in the Cold War since the 1980's.

When the President dies during his term of office, there's a line of succession: Vice President, Speaker, Senior Senator, Cabinet members, etc... The biggest thing is, though: You have to meet the requirements of being a Presidential candidate: Be over 35 years old, be born in the U.S. *natural born citizenship* and have lived in the U.S. for 14 years... To be able to be in the line, you are ineligible if you don't meet them. Madam Albright born in Czech for instance was never part of the line even though she was higher up, nor is a Cabinet member in the current Presidency because she was born in London and also a higher ranking Cabinet member.

With this, a Designated Survivor is chosen every time there's everyone in one place from the Legislative/Executive branch *the Supreme Court/Judicial branch aren't part of the line of succession* who meet the requirements who is brought to an undisclosed location. In case of an attack, they take over the Presidency. There were a lot of people asking what would happen if everyone died in the line of succession, except maybe one who was non-U.S. born, would they still reject that person?

After a bit of googling, I found out what happens is basically if everyone dies, it becomes a State Governmental decision to reoccupy the Senate and from there they elect a President. So I was wondering:

If worse comes to worse, should someone who doesn't meet the requirements of the President of the U.S. but there's no one left in the line of succession who does, should the person who doesn't take over as President?