Commenting because I also have an old chest covered in blankets because of my cat (she likes to sleep there).
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Commenting because I also have an old chest covered in blankets because of my cat (she likes to sleep there).
You could do batting or perhaps even foam to make it more like a seat. I don't think any fabric on it's own is gonna endure kitty claws unless you'd get your hands to something slash resistant like foresters use when they're cutting with chainsaws. Even batting might be too thin unless you stack it up some. One option could be a rug cut in shape or possible EVA foam.
In my head, this photo features 1 cat whose tail wraps all the way around. I'm wondering if a long seat cushion might be the way to go. A cushion whose bottom side has some of that grippy stuff you can get for lining the kitchen shelves/kitchen cabinets. My concern with the "gathered elastic sheet" version is that a good thunk from a jumping cat is likely going to pull the elastic right off. Downside of this idea is that the cats might not be okay with a replacement bed. You could also stuff the cushion with something heavier than just batting, so that it has a little weight to it. Buckwheat hulls, something sorta like that. https://seekatesew.com/how-to-make-a-weighted-blanket/ for some design ideas. I am also wondering about something that is effectively a table runner. Again with something grippy on the downward-facing side. On the top side you could use something heavy duty to protect against the claws - faux leather, duck cloth canvas, upholstery fabrics. Last thought: you could also just put straps diagonally across the corners to hold your piece in place. Then you can use something that is rigid and more likely to hold steady when your cats land all Superman style. Twill or webbing, with or without adjustments like a D ring or velcro or something. Mew!