Jan 3rd is also the date the new House and Senate must be inaugurated and seated.
I bet they will lose the House and see the Senate slip out of their grasp. If and when that happens and there is still no clear winner on the new Potus and VP, then the newly seated House will vote on the new Potus and the Senate will vote on the new VP and they will serve in the interim starting Jan 20. When they finish with all of the wrangling over the votes, the states electoral delegates have to be picked and then the electoral votes are tabulated for the winners of the election.
The electoral votes are approved by the old Senate and the old Senate can disqualify the electoral votes. It wouldn't surprise me if the Senate disqualified a state's electoral votes on the basis of fraud created by mail in voting.
The Senate can completely reject any state whose delegates present two different winners (has happened once), so I would think since that would be clear fraud, any type of fraud should be sufficient cause to deny a state's declared choice.
JM2¢