“The Boltons are planning to betray the Lannisters when they get a chance” Wait, what?

See here and here. Tywin’s original plan always had tension between the interests of the Lannisters and the Boltons:

“Why, do you plan to mistreat her?” His father sounded more curious than concerned. “The girl’s happiness is not my purpose, nor should it be yours. Our alliances in the south may be as solid as Casterly Rock, but there remains the north to win, and the key to the north is Sansa Stark…Come spring, the northmen will have had a bellyful of krakens. When you bring Eddard Stark’s grandson home to claim his birthright, lords and little folk alike will rise as one to place him on the high seat of his ancestors.” (Tyrion III)

Lord Bolton will wed the girl to his bastard son. We shall allow the Dreadfort to fight the ironborn for a few years, and see if he can bring Stark’s other bannermen to heel. Come spring, all of them should be at the end of their strength and ready to bend the knee. The north will go to your son by Sansa Stark … if you ever find enough manhood in you to breed one. Lest you forget, it is not only Joffrey who must needs take a maidenhead.” (Tyrion VI)

“The price was cheap by any measure. The crown shall grant Riverrun to Ser Emmon Frey once the Blackfish yields. Lancel and Daven must marry Frey girls, Joy is to wed one of Lord Walder’s natural sons when she’s old enough, and Roose Bolton becomes Warden of the North and takes home Arya Stark.” (Tyrion VI)

So the Boltons would have the Wardenship and Arya, Tyrion would have Sansa and Winterfell, but that’s not a tenable situation over the long-term because only one of these houses can rule the North. So Tywin wants to wear down Roose Bolton’s power by having him fight the Ironborn and the Stark loyalists, and then turn on him to consolidate power in the person of Tyrion’s Lannister-Stark son. 

Roose Bolton realizes this, but he also realizes that once he’s up in the North with his carefully-hoarded Bolton-Frey army, there’s really nothing the Lannisters can do to him all the way down in King’s Landing:

“Lord Bolton aspires to more than mere lordship. Why not King of the North? Tywin Lannister is dead, the Kingslayer is maimed, the Imp is fled. The Lannisters are a spent force, and you were kind enough to rid him of the Starks. Old Walder Frey will not object to his fat little Walda becoming a queen. White Harbor might prove troublesome should Lord Wyman survive this coming battle … but I am quite sure that he will not. No more than Stannis. Roose will remove both of them, as he removed the Young Wolf. Who else is there?” (ADWD, Prince of Winterfell)