Is Mass Converted to Energy in Fusion/Fission of Atoms?

According the Einstein's $E=mc^2$ the energy released in fusion/fission of atoms corresponds to a decrease of mass. The atoms before fusion/fission have more mass than after, with the mass difference being transformed to energy. 

The process of fusion/fisson of atoms is thus considered to be different from chemical reactions generating energy from recombination of the electronic clouds of molecules/atoms/ions into new energy levels in exotermic reactions producing energy and in endothermic reactions consuming energy. Her the total mass of the molcules/atoms/ions involved does not change, only the total electronic energy level, up or down. 

A molecule/atom/ion is here considered to be one or more positively charged kernels surrounded by negatively charged electrons held together by electromagnetic Coulomb forces with corresponding electronic energies.  

In the same way atom kernels are viewed to consist of collections of protons and neutrons held together by the strong force overriding electromagnetic repulsion between protons with corresponding nuclear energies. In fusion/fission atomic kernels recombine just like electrons clouds of molecules/atoms/ions into new energy levels with corresponding production/consumption of energy. Does here the total mass change?  

In modern physics it does because energy is viewed to be equivalent to mass according to Einstein's $E=mc^2$. 

In classical physics energy mass is inertial/gravitational mass and so is not expected to change under recombination of molecules/atoms/ions, because a sufficient reason in the spirit of Leibniz is lacking. Recombination of electron configurations into new electronic energy levels is enough. This is what is observed. Reorganising a truck load does not change its mass/weight, but possibly its potential energy by putting things on top of each other.  

Why expect anything different when reorganising collections of protons and neutrons with corresponding change of nuclear energy?  Is there any sufficient reason? If not, then what?

But what about experiments? Isn't it true that 0.1% mass is lost in fission energy? Maybe, but measurement of inertial/gravitational mass is very delicate. How do you measure the loss of mass in an atomic bomb explosion? 

The fission mass of Little Boy was about 1 kilo releasing about $64\times 10^{12}$ Joule, to be compared with the $10^{17}$ Joule of the SI standard based on Einstein's $E=mc^2$ discussed in the previous post, with thus a formal loss of mass of less than 0.1%. Was it measured?

We may compare with the discussion of phlogiston theory predicting a loss of mass in chemical reactions into phlogistons of energy, which showed to be wrong because mass showed to be conserved.