Hi blooming,
Coconut fibre liners have physical properties that suit basketry; it combines oxygen circulation, moisture retention, adequate drainage & bland looks.
Peat moss will not hold the shape or be a struggle to fill; paper covers a lot of kinds.
Burlap is porous for air circulation & drainage. You can hot glue it to the basket rim & fold it down in. It will rot out faster than coconut & of course cocoa is the source of chocolate.
Nursery shade cloth is synthetic, it won't rot out readily; black common & inelegant.
Embroidery sheets, the white backing you work threads into, will have strength, breathe & drain.