Is there anything growing there now? It would help to identify whether weeds are are annuals or perennials. Some are persistent and will re-grow from roots. Or maybe you have already herbicided...?
I think digging a trench all around the perimeter for deep filling is a good idea. I also think you could get away with NOT removing the dirt from the central portion -- maybe only an inch or so -- then laying down soaked down cardboard (thick flattened corrugated boxes -- you can get them from businesses if you don't have any) and covering with just enough mulch to hide the cardboard and make it look nice. I would mold the edges of the wet cardboard down into the trench/moat to keep weeds from growing and help hold down the cardboard.
I don't recommend weedblock/landscape fabric because they become more of an obstacle down the line. Cardboard will break down, but you can dig or use machinery/power tools as necessary to redo the area as needed.
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