We have Mulberry trees that we inherited when we bought the property. They grow like weeds, literally and are doing well despite crowding from other trees. Some are more shaded than others but that doesn't seem to bother them. I like them well enough on the back borders but not by the house or drive where we walk daily. The fruit makes a mess, gets tracked everywhere and stains. Ours were 20+ feet when we got the house but a friend of mine prunes her Mulberry and successfully uses the fruit in to make jam.

I imagine pawpaws and elderberry would work well as well as mulberry. But the real winner would be an evergreen huckleberry assuming the pH is around 5.5 or lower. Heavy mulch.

My shade tolerant list: (Raintree Nursery)
Partial Shade
May have less fruit.

Alpine Strawberry
Bamboo (some types)
Dogwood
Elderberry
Evergreen & Red Huckleberry
Filbert
Gooseberry
Hydrangea
Jasmine
Lipstick Strawberry
Lingonberry
Magnolia Vines
Musk Strawberry
Paw Paw
Pie Cherry
Salmonberry
Serviceberry
Sugar Maple
Thimbleberry
blackberry

Full Shade

Bamboo (in sunny climates)
Currants
Carolina Allspice
Evergreen Huckleberry
Flowering Currant
Wasabi
Highbush Cranberry
blackberry

Based on the wild persimmon trees I've observed on my place I think a persimmon would be very happy in the situation you describe. However, persimmons seem to be slower growing trees to start with, and I think limited sunlight and root competition would slow them down further, so although I think they'd do very well, I'd guess they'd take a long time to size up and reach fruiting age.

Have you thought about nut trees? I'm wondering particularly about chestnuts... maybe also a grafted hickory. Would squirrels be too much of a problem with the adjacent woods? Would they be able to jump directly from the adjacent woods into the branches of the tree once it matured?