I think if you got hold of a bunch of pea sticks with lots of twiggy bits on them (hazel if you can get hold of any) and inserted these right round the veg bed so the twiggy bits touch, that might deter the pets a bit. They'd need to be at least 2-3 ft high, but you'd be able to pull one or two out for weeding or harvesting. Otherwise, what about some of the lightweight expandable trellis, tied on to small posts around the edge?
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I planted catmint once for my two to enjoy. Problem was it attracted every Tom, Dick and Harry into my garden too 😬.