Reddit reviews Heavy duty eyelet plier and 100 eyelets for tarpaulins/sheets/covers TE446

for something like canvas versus the weight of a cat i think you'd probably get away with double folding the ends and some tough stitching to make little segments for the screws to pass through- the stress is kind of spread the width of the fabric if it's clamped tight...

but you could stick some tarp/canvas eyelets in around where the screws or dowels or whatever pass through, then counter sink the wood around the holes a little so it still fits flush.