Assuming your fuyu is grafted on American persimmon root stock they are extremely tough trees with very few diseases or pests that bother them. Don't over think it, plant the plants you want around it. Paw paw, oak, black locust, ash, pine, blueberries, elderberry, Jerusalem artichoke, clover, millet, crabapple, stag horn sumac, and devils walking stick are all things that play well with persimmon on our property.
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Yes - Japanese persimmons are wonderful. I grow native coral beans beneath them as a nitrogen fixer, along with sweet potatoes as a ground cover. I also grow Tithonia diversifolia next to them as a chop 'n' drop phosphorous accumulator. Herbs, wildflowers, ornamentals and other bits and pieces get tucked around them too. No matter what I plant, they do fine.