He's pretty big for a lhw. Tall lhw's like Michael Spinks and Bob Foster eventually moved up. But it would be best for Dawson to stay as a LHW as long as possible. The problem with Dawson is that he's been picky about which opponent he wants to fight for the last three years, which is why he hasn't even come close to cleaning out the division despite being much more talented than the rest of the competition. Dawson will stay as a LHW as long as he can, if Tarver can do it, then so can Dawson.
Chad needs to build up his upperbody strength before moving to CW or HW guys there weigh more so naturally would be tougher to pick off their feet.
I don't see Chad Dawson having success at heavy, unless it's a late run to conclude his career like RJJ taking on a paper champ like Ruiz. He will be a cruiserweight champion one day though--he's got enough size for that division.