For what it's worth, I've since established that the low order bit corruption only occurs when the input and output texture formats are not identical (in terms of their number of channels).
For instance, the original test program copied the first three components of an RGBA texture to an RGB texture. If it's instead set up to copy an RGBA texture to an RGBA texture, then there is no low order bit corruption -- likewise for RGB to RGB and Alpha to Alpha.
So, I am now able to temporarily work around the problem in my real program, although the 'solution' is not optimal since it involves eating up some extra time and space because I must add in an extra pixel component that is never ever used.