The design seems incredibly complex to manufacture, and would need to outperform existing munitions to be viable in your story without suspension of disbelief.
How do you disengage the ballistic shell? Removing it will likely render your bullet horribly innaccurate.
With small, thin (easily deformed) hooks and a hard 'penetrating' material, how does it offer expansion and soft-tissue deceleration beyond existing hollow-points?
With a 'free floating' firing pin, how do you guarantee the safety of the user from accidental ignition during loading/handling/transport?
I have a few other questions, but I don't actually need any of them answered... I'm just tossing out hurdles which would prevent this kind of ammunition being a viable replacement for what already exists. Frankly, if you're trying to make things burn and pierce, I would have thought you'd be looking at existing depleted-uranium rounds, and then modifying their design to your needs. If you're trying to pierce and expand within the body, then you're in the market for a kind of bullet with cross-purposes. You can't have a soft shell and a hard center if you want to penetrate, and you can't have a hard shell and a soft center if you want to expand. You need a way to shed the hard shell of the bullet after penetration, but before exiting the body... and that's one hell of a challenge.
Might I just suggest trying to find something which fragments really well, pierces moderately, and is highly toxic or corrosive?
Or maybe look into frozen bullets... Not water though. Maybe mercury?
If this is a vein of thought which interests you, it might be worth considering a propulsion method which doesn't generate heat. Compressed air would be my best guess.
Speaking as an avid reader of all things sci-fi, my view is that it's simpler to accept something into the flow of the story if it's easily explained, rather than something which really needs a diagram to get across.