If the size specifications are correct, then this 50L bag, for example, can be placed inside: Zeller present laundry bin But it has NOT been tested! In general, the bin can be lined with some kind of fabric bag.
Or you can quickly sew something yourself. Put the bucket on the fabric - mark the curve - cut it out - cut the side wall and sew it to the curve - if necessary fold it over at the top, sew it and pull in an elastic band. 20min work or so :-)

This laundry bag does not come with a separate laundry bag and unfortunately there is no matching one.