My understanding is that the 20th Amendment applies AFTER Obama would become “President-elect” which means AFTER the Electoral College votes on December 15 and if they vote for him. If Obama is found ineligible to hold office before the Electoral College meets, the Electors are free to vote for someone else and have the duty to only vote for someone who qualifies constitutionally.
I believe it is strongly possible that if SCOTUS rules on Obama being ineligible and the DNC instructs the electors to vote for him anyway, the Congress would certify the results claiming they have the final say on eligibility.
The 20th Amendment applies regardless. Whether the electors are free to vote otherwise should Obama be found to be ineligible depends in part on state laws. I don't see that happening by the 15th. The Berg case isn't going to move that fast, and even if the court agrees to hear the case being reviewed on Friday then there still isn't enough time to schedule a full blown hearing. If Obama is found to be ineligible then it'll happen after that.