He wont try to avenge Cat purposefully because Littlefinger isn't stupid. He probably will fight with the Freys, but not for Cat, simply as a means to achieve further political control.
Also the entire Frey situtation is probably about to get really, really complicated. Since Walder is likely to die within WoW and Edwyn Frey (rightful heir of Walder) and Black Walder (second in line of the succession) are most likely going to fight a succession war. Littlefinger is smart enough to exploit the situation, I would guess. Though we wouldn't know the details until WoW comes out.
Him running off with Sansa and turning her from high borne noble to peasant bastard could be just another way of getting back and Cat and the Starks for considering him lowborn scum.
Is marrying her to a relatively handsome, charming young lord with a powerful army and giving her a political position where she can avenge her family also a way of getting back at Cat? He hid her as a bastard because she needed to keep her identity hidden. Elsewise his plot would've failed and she'd never get to be Queen of the Vale.