I still don't think you would notice a big change when looking at 3/8" of a difference. The key here, I think, is the wood stiffness associated with thickness, not the weight. For instance, if you take a guitar with, say, a 1" thick body, and compare it to a 2" thick body, the 2" thick body will have more sustain because the body isn't able to flex as much as the 1" body does, thus it won't dampen as much of the vibrations.
So, will you see a difference between a 2" body and a 1 5/8" body? Maybe, but only if the wood is relatively soft. Think of it this way: the effect of the tension of the strings (which would affect how much flex the body is under) will only be countered so much as the thickness increases. Its the idea of diminishing returns in economics: for every 1/8" extra thickness added, you get less sustain than the previous 1/8" added. You can only increase the stiffness of the body to a certain point. Beyond that point, the returns are negligible.