Nope, not a particularly crazy idea. I wouldn't consider the Pearsons, Hunters, or C&Cs as particularly good bluewater boats. I wouldn't necessarily say you need to have a bluewater capable boat to do the trip down the east coast and out along the islands to the Caribbean either. However, if you're planning to cross the Atlantic, it might be wiser to get a true bluewater capable boat.
However, finding a boat with two heads, a separate shower, and two private cabins for your budget is going to be a bit tough, especially since I recommend that you reserve about 15-20% of your boat buying budget for refitting, repairing, and upgrading any boat you do buy.
I would recommend that you get a sloop-rigged boat about 30-35' in LOA since you will be single-handing much of the time. Finding a boat with solar, wind and generator power in that size range highly unlikely. Gensets aren't very common on boats, and even less so on boats <40' in size, since they take a fair amount of space to install. Dual heads are also pretty uncommon on boats < 40'.
A liveable galley is usually a serious compromise on most monohulls. The galleys on catamarans are generally much more spacious and more than just functional, which is how I would describe the galleys on most monohulls.
BTW, most bluewater boats are going to have less room than the coastal cruising boats are, since that is typically how bluewater boats are designed. They often have narrower beams, and smaller salons and galleys, which make them safer in heavy weather, but somewhat less suitable for living aboard. However, at least two couples were living aboard Alberg 30s with a child for some time... one couple still does and has a blog about it.
One boat that might be a good choice for you is the Hallberg Rassy 35
. There are several for sale on Boats.com
. Another good candidate might be the Southern Cross 35. Its smaller siblings, the Southern Cross 28 and 31 have both made circumnavigations. Donna Lange just finished a circumnavigation in an SC28 earlier this year. One caveat on the SC boats are that they have an Airex-cored hull.... so need to be checked for water intrusion into the hull.