Should I mention a published paper in a disreputable journal in my resumé for admission to a PhD scholarship?
I understand your conflict.  On the one hand, you have produced a manuscript detailing your hard work.  On the other hand, you don't want to appear associated with a fake/predatory/otherwise questionable journal.

My suggestion is that you do include this work in some form in your CV. For example, you might have a few sections:

1) Peer-reviewed publications

2) Publications in-press

3) Other work / manuscripts / manuscripts in progress, etc.

If it were me, I might consider including this paper under #3, potentially withholding the journal name if I felt so strongly as to question whether it might even be called a journal.  Alternatively, you might include a note underneath that line item stating "submitted to non-peer reviewed journal without author's knowledge".  While this would be fairly strange to see, it would at least offer a discussion point, while giving you credit for work performed.