I think, and this is based more on TV clues than book ones, that Littlefinger wants what the Spider wants, a kingdom not at war, with a ruler who will help the people, not just worry about their own power. I *think* Littlefinger is preparing Westeros for the eventual arrival of Danny.
“Eternity is a terrible thought. I mean, where's it going to end?”
― Tom Stoppard, Rosencrantz and Guildenstern Are Dead
― Tom Stoppard, Rosencrantz and Guildenstern Are Dead