Oh I don't know, once Littlefinger secured the lordship of the Vale he had a pretty decent opportunity to just carry out Lysa Arryn's plan: hole up in the Eyrie until it all blows over. He might have even been able to hold off the army of the dead (surviving on the grain he bought up in anticipation of the long winter). The only thing that could have dislodged him from the Eyrie would have been Dany's dragons, but it's unlikely she'd bother attacking the Vale without first defeating the Lannisters and the Night King (and she probably can't do both alone).

Of course, for Littlefinger to take this path he'd need to take the threat of the White Walkers seriously, and to have a better grip on his limits. In this alternate timeline, Lord Baelish locks his gaze with the Night King for a full generation, growing old as the hungerless army of the dead lays siege to the Eyrie. Eventually, the long night ends, and the army of the dead recedes, leaving behind a depopulated Westeros free for the taking. Baelish rides south to King's Landing, finding it abandoned, and at last claims the Iron Throne. King Petr would spend his final years overseeing the reconstruction of Westeros. He leaves no heirs, and invests no new royal houses. Instead, he declares himself The Last King, Wise and Eternal Ruler of Westeros from this day until the end of days. He commissions 'The Commonwealth of Westeros' to rule the land - a government with run by elected ministers fully capable of functioning after his death. So begins the eternal rule of King Petr the Patient.