Royal Prerogative - One of the reasons I think the throne track gets neglected is that despite the potential benefits of tie-breaking, it can be a toss up whether it is actually an advantage to move first or last in a given round. So there isn't a huge incentive to be sure you move first, generally. To counteract this, the Iron Throne token could be used once per turn by the King to skip his turn on the track order, allowing them to save a march for a counterattack, or perhaps a raid until its confirmed that an ally has kept faith with their own moves.
The Kingsguard - The other route would be to allow the King to deploy the Kingsguard to one battle per turn, adding 1 Sword and 1 Fortification icon to their house card before the cards are revealed.
I think with either change, you would see more competition for the Throne in the bidding, and potentially have it drawing more focus from Doran. If anyone wants to suggest other options, and/or try out one variant or another in their home game and report back, please do.