Uses for tzimbalo fruit of varying ripeness?
When it comes to the part about underripe fruit: that's ok to use in chutney, pickles, and other preserved recipes. There are recipes which are made intentionally for underripe fruit, like mangoes. You may want to adjust the sugar to your taste. 

As for other uses, while I've never had tzimbalis, there are several methods typically used for very tangy fruit. Pickled pieces of it (alone or as a mixed pickle) and liqueur come to mind. Aside of that, you could try pureeing it and see if it is a good substitute for tamarind paste. Or mix the puree with random creamy ingredients and see which combination strikes you as a good dip. 

Also, these being a solanum, try them as a substitute in recipes for unripe tomatoes. Although these will have a high overlap with the pickle suggestion, you might find a salsa or two that work.