In any case, we have to go back to the question that the GP asked, that dogs are malleable doesn't help us. Does reduced child mortality really cause more mutations to accumulate?
However, dog breeds are not species and, given their non-random mating, don't really tell us much about whether 200 years is long in vertebrate evolutionary time.
As far as humans go, I will grant that allele frequencies may have changed if you pick an isolated subpopulation of humans (say, everyone in Greenland) but even that seems extremely unlikely.
As an aside, there are researchers who are using the caste system in India as a way to explore genetic architectures within a civilization. As these castes have existed for considerably longer than the period we're discussing, meaningful allele frequency changes may arise.