Eliea and Abssinth, I like both your ideas for the lining. Unfortunately, I don't have the necessary supplies and it's too cold to go out today. I try to hibernate as much as possible on the days I don't have to go to work.
So, I tried to come up with something that wouldn't require snaps or velcro or zippers since I don't have any. Here's what I came up with:
Cut a piece of fabric the width of the container and twice the length. Sew into tube (the fit should be as snug as possible), stitch across the bottom, then miter the corners. Finish the top edge.
Turn tube wrong side out then place in container.
Pull top of tube down the outside of the container. May require some effort since the tube needs to fit snugly.
Inside of lined container. Bottom is left uncovered.
Place lined container inside bag. If the lining is snug enough, it should stay in place while the bag is being pulled up.
Finished lined bucket tote.