Q. Does Chicago prefer “whosever” or “whoever’s”?
A. We’d follow the advice in Garner’s Modern English Usage, 5th ed. (2022): “Whosever is the traditionally correct form, but it’s very much on the wane. Whoever’s is now the preferred colloquial form” (see the entry “whoever; whomever: B. Possessive Forms”).
In other words, practically nobody uses whosever anymore, so you’re better off with whoever’s, at least in contexts that are less than formal. For example, Whoever’s [not Whosever] car is parked on the sidewalk needs to move it.
If the context is formal, however, you should probably avoid this awkward construction. Whoever is responsible for editing your prose [not Whoever’s job it is to edit your prose, and not Whosever job it is to edit your prose] will thank you, whoever you might be.