Can I properly install Roxul mineral wool batts in a mostly-drywalled stud bay by pushing them up from below?
Roxul batts do not compress a lot, well not nearly as much as traditional fiberglass.  They are also pretty rigid.  They can definitely go 6 feet or so in a wall cavity.  I have used them to do what you are doing, when there aren't things in the way.

I am guessing you believe you don't have cross bracing or electric/plumbing or obviously this won't work.  I would unroll a batt, cut to height size (possibly cut to width), push up slowly starting at top of batt.  If it snags and is resisting I would cut it.  Then push up with broom or something similar and just keep on going until you fill it.  Even if the Roxul slides a little you can stuff the bottom to hold it.  The stuff really compacts well and holds its place.  Why I tend to use it in basement joists.

Also if you place a small screw in the top side of your batt (just push it in so it isn't hanging out) you can use a magnetic stud finder to see where you are at.