Could you pick up short rows around the top between the points where they cross over in the back and knit them up to go higher up in the neck area?

No. It's a wrap. It's a really long scarf with cuffs on each end. So you put on one cuff, wrap around the entire body and then stick your other arm in the other cuff.

I guess if you don’t want to fix the draping in place permanently by knitting around the gap, you could knit a dickie (with a turtleneck?) in the same yarn to go under it.

Someone suggested I rip it apart and make a real sweater instead. Considering I'll probably never wear it, I may just do that.