What size is the container?

8 inches deep 8 inches wide

To me, the fact that it recovers overnight suggests it's getting too much heat or not enough water. But I'm not an expert.

When you're growing in dark-colored containers in direct sun, there can be a problem with the soil heating too much. So if that's the case here, you might cover the pot in something that will absorb less heat from the sun: for instance, a white plastic bag, a piece of white cloth, or a sheet of silver mylar (space blanket stuff).

If you don't get more answers here, you might consider cross-posting on the Container Gardening forum.

I have wild ground cherries, but I've never planted them, let alone indoors.

The flowers dropping might be related to whatever is making the plants droop, or could be a pollination issue. According to the California Rare Fruit Growers, "Cape gooseberries are self-pollinated but pollination is enhanced by a gentle shaking of the flowering stems or giving the plants a light spraying with water." Shaking the stems a bit is also recommended for indoor-grown tomatoes.

Thank you for your informed answer. I don't think it's lack of water.. at the moment its roots don't cover much of the pot and I only need to water is every other week.. though the time between waterings is geting shorter.

It's in a light brown pot.. so I think you could be right about the roots getting too hot in the soil. I'll cover it and see what hapens

I have the same phenomenon with a very large pot. On a hot day it wilts, as do the artichokes. Perhaps they like coolish weather like in the Andes