I do realize that the possessive was technically incorrect. However, the spelling 'wrap' has been used colloquially for quite some time.... my family's been doing so for at least twenty years, actually. I would assume that 'bum wrap' refers to something that is 'packaged' in such a way as to conceal what is actually inside, like when you place a tiny jewelry box in several larger ones.... or even when a person is careless and tears a bit of the package or whatever is going into it. In this case, the 'packaging' is with words.... very similar to the context of a 'bum rap'.... which, as you pointed out, means something like 'bad talking' or even 'bad rep'.

Just my two cents...