Hmm... I would just rake off the mulch, mark off the area by edge cutting the grass along the perimeter BUT NOT DIGGING/LIFTING THE GRASS (or maybe remove just a 4" wide band all the way around), put down some unfinished compost (nothing recognizable but chunky and still fair amount of earthworms present), water well, then wet cardboard the area and mulch with whatever I can enough to cover and hide the cardboard, then decorative mulch if I'm so inclined with some kind of uniform mulch. Grass/sod should die and break down underneath the cardboard.
...just yesterday watched my neighbor's kid and neighbor laboriously digging out the grass/sod around their front yard tree. The teenager was REALLY digging into the ground and I was thinking yikes Maple tree is shallow rooted.... Then they wheelbarrowed all that rich sod off to the woods, THEN spread bags of purchased tops soil and mulch. For all intents and purposes, it looked very nice afterwards.
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