Raspberries

If they are Autumn fruiting, they don't particularly need to be in a cage. Uncaged, some would be got by birds but not too many. You could have a few in a tub in the cage if you want some Summer fruiting ones and are able to water when dry. There are a some dwarfs on the market but no idea what they are like. I've seen Ruby, Yummy and sweet sunshine.
I've got a small clump of autumn fruiting Joan J that escaped from a cage and I've had lots of fruit off it. The birds don't bother with it later in the year.

Edit: But I would probably do as you suggest: cosset them while they are producing enough fruit.