No, it's not unknown. The answer is no.
".. no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States".
Serving two terms means Clinton would not be eligible for VP either due to the proximity of being a third term President. This one is very easy.
Also, Clinton was very overrated. I laugh every time I hear about how good for the federal deficit he was -- only from Clinton fanatics or people who don't understand how the deficit works.
Actually, no, it's not an easy answer. It's only easy if you want it to be easy, however, easy doesn't make a matter 'legally settled'.
The 12 Amendment says that no one can become vice president if they are "ineligible to the office of the presidency....and one of those ineligibilities is mentioned in the 22nd Amendment, which clearly says that the person cannot "be elected".
Well, a person appointed by a sitting President, who was once a 2-term President, isn't 'elected'....and therefore, technically, doesn't meet the constitutional ineligibility criteria, which is something that could at least be argued in court.
If the issue was that if something were to happen to the POTUS, then the VPOTUS may or may not, in this situation, be allowed to 'switch seats'.
Again, the larger point is that this is not a constitutionally settled matter from the perspective of the USSC, which is the only body that can settle constitutional matters,.to some theoretical open loopholes and technicalities that could be exploited between the arguable daylight between the 12th and 22nd Amendments.
Do I ever think it would happen? No. Do I think it could be argued in court? Yes.
And i'm not sure why you brought up Clinton, since I didn't mention him at all. If you wanted to talk about laughable 2 term presidents, I can produce some real knee-slappers from the most recent republican 2 term chimp-in-chief.