If she doesn't like the Wal-mart bag, then try a Macy's bag :D . Sorry, just couldn't resist. And thanks for the naming a practical use for the front arm rest, it's a poor excuse for an actual arm rest, in my book.
Getting back to your question, my personal view is that there is no classy way to handle large pieces of garbage, the Wal-mart bag solution is as good as any, but I'd put it on the floor in the back seat so it'll be out of the way (works only if the back seats are usually unoccupied).
Or you can do what I do, I avoid eating or drinking in the car so the only refuse I get are small stuff like gas receipts and an occasional gum wrapper, I put them in the center console or door pocket till I get to my destination then I take it with me.
←
yup, I'm with you on that one!