The Senate has no right to make up their own rules about who may or may not be a Senator. While the Senate is a Federal body, the elections and appointments to the Senate are strictly the right of the individual states. The only qualifications the Senate is allowed to bar a person for are not meeting the very short list in the US Constitution:
1. At least 25 years old
2. A Citizen of the United States for at least seven years
3. At the time of election, must be a legal resident of the state in which he has been chosen
That's the list. A sitting Senator may be expelled, but it takes a 2/3 majority vote to do so.