Hoster Tully does agree to send aid to the North.
It turned out not to be as hard as Zakary was expecting. He really does feel guilty about the suffering he's put his daughters through, and reaches for the hope of using that to mend his relationship with Anavett like he's drowning and Zak offered him his only lifeline. Zak feels a little bit guilty about playing the man like that, but on the other hand he does not lie about anything, at least not directly, plus Tully really has dug his own grave there. And what really matters is that the people of the realm will be safer, even if it's not the people of the realm under Tully's personal care.
And beyond that, war preparation proceeds apace. Zak organises their efforts efficiently and, to all appearances, almost effortlessly; he has a constantly-updating mental map of their resources and where and when and how they'll be deployed, and for any questions or concerns he can't immediately address he knows whom to ask or what to do, off the top of his head. It's energising and motivating, to see the head of the rebellion looking to confident and self-assured all the time. Even those bannermen of his who had originally resisted him come to grudgingly respect his competence, and it's easy to believe everything will work out, with him at the helm.