I always try and avoid modifying plugins so I can get updates.
I was able to solve it a different way, though it’s not very pretty either. I used the built-in Custom Menu widget and using CSS hid the items I didn’t want to show. The Custom Menu widget shows the entire menu hierarchy, so you can select items that aren’t related to the current page/post and hide them via CSS. In the end it produces the same results as Advanced Menu Widget, but will also show the proper items on a post.
I’m not thrilled that there is HTML rendering for all the menu items that I’m hiding, but this will work well enough for now.