If Littlefinger had waited things out, whether the Boltons or Stannis come out on top he'd be left out in the cold and the Lannisters would still be Lannistering. Forcing a conflict between the Lannisters and Boltons was what he needed to do to move up -- investing in the Boltons was just an aspect of that.
And yeah, he could've just lied to Cersei, but then like...
*Lannister army shows up at Winterfell*
"We demand you hand over the traitor Sansa Stark!"
"What?"
"Sansa Stark! We know she married Ramsay!"
"What? What are you talking about? Literally ask anyone in the North, that never happened, we don't have Sansa, come inside and look."
"Oh, okay. Welp, see ya later!"