The first thing you need to do is determine if your system is setup for DirecTV or Dish. Call King Dome and give them the model number. They can tell you what you are currently set for and probably can help you change it if you prefer the other network. Your King Dome cannot receive HD on DirecTV but it may be capable of being programmed for HD on Dish.
No dome system can receive DirecTV HD, but domes can be used for Dish HD. However, a limitation of all dome systems is that they can only receive from one satellite at a time and, therefore, are of limited use with DVRs if you wish to record one channel while watching another since they can't do that unless both channels happen to be on the same satellite.
Having owned a King Dome vintage 2005 I can say I was never impressed with its performance. Don't get discouraged by satellite TV if it doesn't work as well as you hoped. My suggestion is use it on your trip; you can always supplement it with one of the "carry-out" systems which work quite well or you eventually can replace it by a rooftop unit. Just start out with the network you prefer, you can always change receivers and/or upgrade to HD at a later date.
