Dave is ready to play distraction, he can attack from afar. LET'S DEFFEND ALL THEIR HARD WORK. Also you know he's also neutral and even if he can't see through Sansa's lies, he's seen enough (plus what Jon has told him) to swear to her if needed, though he may want to negotiate a word or two according to what it says. ;)

I'll wait for the time-skip calendar to come up just in case, but it's highly likely he'll fall asleep in the Coth.