Just so that we're clear, Ser Gregor's Band get +1 STR for each card that isn't a House card rather than for each card that isn't affiliated with its owner's house. Hope the distinction is clear. For example, if I my house is Lannister and I get a point of power on Cersei then that counts as +1 STR since Cersei is a Character card and not a House card, even though she is a card affiliated to my house.
Does that sound right?