The upper unit whether Coleman(RVP) or Dometic, the type inside tstat has NO bearing on the replacement. As long as you replace with the same brand as the upper unit. The ONLY exception is the new style Dometic Penguin AC/HP. The upper unit has the 120 multiplug that connects to the upper control board. The upper control board will stay with the new unit. The upper control board is just 1 of 3 components.
1. Top AC unit
2. Upper control Tstat box
3. The wall tstat
Last, there is no remote "bing" type RV tstat. There have been threads on making a tstat work on RV's but the cost was Hundreds of dollars and electronic knowledge to do. Doug