The pictures appear to be at least two different BCs, with both the rear dump and the inflator elbow broken. By my count, that's three broken valves. Is this a common issue for you? How hard are you pulling on the dump?
Seriously though, how did all of these break? If it's abuse, then that's its own story. I'd be comfortable repairing something if it was accidentally broken by an overzealous dive buddy, or a garage accident or something. If it's just due to age and fatigue, I'd personally buy a new BC rather than risk other failures that may be just barely holding together, especially if you're not sure how to test a repaired dump valve. My intention isn't to be rude with that statement, just pointing out a piece of equipment that's seen one failure due to age or general fatigue is likely not far from seeing more failures for the same reason. If you're not able to confidently inspect for "looks suspect," then it's time to junk it and start fresh. Finding something that already failed is easy. Finding something that's
about to fail is more challenging, and arguably more critical.
Edit: typo