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.
If I understand your question, the answer is "yes". That is, mass and energy remain constant because not very much mass is "converted". The accounting of energy typically includes "potential" energy. This is a way of treating the energy as if it had not been converted into mass. Since the amount of energy is low, the mass change is extremely tiny and can be ignored. The potential energy term keeps the energy calculation in balance.
In most cases, "binding" energy is negative. That means that energy was given off during the binding and is missing from the bound particle. The potential energy of the bound particle is then negative. The bound particle will, I believe, have a mass which exactly accounts for this missing energy.