Can a subordinate clause have two functions at the same time?

What can we do that is more important than choosing the men and women who serve us by running our government?

You're right, the portion in bold is a relative clause.

that is more important than choosing the men and women who serve us by running our government?

This cannot be a complement because no linking verb could directly precede it. You could not write, "What is that is more important than choosing the men..."

that is more important than choosing the men and women who serve us by running our government?

The portion in bold is the complement of the subject "that" of the relative clause.

What is more important than choosing the men and women who serve us by running our government?

However, if by "slight rewording" includes deleting the subject "that" from the relative clause, then it can become a complement as shown here in bold, which is the same complement as further up in the relative clause further up. It simply moves from being a complement of a relative clause to being a complement of a sentence.