I don't mean this as anything but advice, but... things are much different for you here, than in your world. You do not currently have an army at your disposal, and there are rulers of equal power spread across the lands.
In saying that... I don't think it would be wise to pressure the Starks to bend the knee. The land we are being given for New Westeros may not be as massive as your world, but it's still big enough to surely come to an agreement of some sort. Allow them Winterfell and the other lands of the north, while you would rule all the lands of the south.
I feel this would avoid any conflict that we, frankly, don't need and aren't at all prepared for.