Some ideas of plants I know of that can grow in partial shade that have berries or small berry like fruit include:
Arctic Beauty Kiwi (grape sized kiwi)
Salal
Wintergreen
Thimble berries
Gooseberries
Currants
Salmon berries
Alpine Strawberries
Musk Strawberries

I am sure that there are others... but these are ones that I have been able to successfully grow in partial shade conditions.

Don't know if this will work for you, but I have both red raspberries and a fig on the east side of my house. They get full morning sun, but after solar midday, are in house shade because they are right up against the building. Both seem to do fine.

I don't know about domestic varieties, but the biggest, brightest, wild blackberries on my property only get a few hours of morning sun.

The ones in patches that receive all-day sun, while still tasty, always seem to be much much smaller.