Plasma TV's are possibly the biggest source of radio frequency interference in a house. I don't know how it would affect anything in the 1090 MHz range and it may not but when ours is on, I cannot use my AOR or Icoms even up into UHF and VHF without serious interference across the bands (despite outdoor antennas and being nowhere near the TV in the house). No level of power filtering or screening on the TV or receivers has made any difference.
In fact it is so bad I actually disconnect my receivers when it is on to prevent any chance of overloading them.
I definitely would not place the Radar Box with the antenna anywhere close to the plasma until you are sure it won't interfere; if running an outdoor antenna you may be fine.
Many plasmas have USB ports with alternate display modes for attaching computers, video cams or viewing pictures from memory sticks. If yours does not search for "video USB adapter" on Google - Amazon alone has a ton of them.
I don't know if they would work with Radar Box as they are made for mini-cams and cameras mostly but you may look at specs and see.
