Not a lawfag so maybe a lawfag can clarify this but…
US Constitution A2,S2,P2
In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party,the supreme Court shall have original Jurisdiction.
So if I am reading that correctly Trump really has to wait to go to SCOTUS until the vote has been certified, because until then the 'states' haven't acted and so they aren't a party to the action, only individual officials are. Until the states act and certify the vote SCOTUS only has appellate jurisdiction so any claims have to work their way through the appeals process. Once certification occurs Trump can file directly with SCOTUS.
Put another way what we are seeing right now is only skirmishes while the main forces are arranging themselves. The real battle starts after certification. Then a claim can be filed with SCOTUS, and the actual evidence can be presented.