Hmm...
I tried to side step this a little, hoping to keep the picture simpler.
In the case of a helium atom, the binding energy is negative. That is one way of saying that energy is released during the process of binding. That is why one can use "fusion" as a source of energy. That energy has to be added back in order to "un-fuse" the atom, so the separated particles end up having more mass than the original helium atom.
Fission of uranium exhibits the opposite case. It takes just a little energy to cause the uranium atom to separate into smaller particles. As the separation occurs, much stored energy is released. The mass of all the particles at the end is less than the mass of the original atom.
The fission example is like a compressed spring with a string tied to keep the spring compressed. If a little energy is added to break the string, the spring can release its stored energy.
Now you've lost me, though your prolly didn't know you ever had me. lol
What would produce (release) energy from this atom at the time it breaks up? I mean, the way your describe it makes it look like a basic chemical reaction, such as combustion would be an energy user, not energy producer. Aren't chemical reactions mostly conversion of one kind of energy into another kind, so there's not actually much, if any involvement of conversion of mass into energy involved? I thought combustion converts potential energy (stored kinetic energy?) into heat & light, no? (Am I dating myself & my education by saying this? lol) Maybe things have changed, but I didn't think there was much change, if any on the atomic level & it happens on the molecular level in chemical reactions. I know E=mc^2 still applies to chemical reactions, but I thought it mostly explained why mass & energy remain constant.