I'm inclined to believe the doctor; I think he's just really, really, really, really stupid. Like dumber than Bush II.
But as to the weight: grasping at straws, dude. Really. 239 lbs isn't that bad of a number, looking at him that's what I'd guess, and I'm pretty good at estimates, if I may say so.
But even still: you CANNOT estimate ones weight with any degree of certainty. A huge amount of your weight is in your bones, their size and density. There's a comfortable 35 lb swing for bone mass, and we know the president is not known for exercise, a density building activity, which should put his bone density FAR below average.
TL;DR: too much of your weight is invisible to ever really complain about the accuracy; you gotta take it on faith.