Is DNA mutation locally energetically stabilizing the DNA molecule
a spontaneous mutation (seen as a chemical transformation) should lower the energy of the system


Why do you think that? Because it’s an endothermic reaction?

Consider that mutations don’t happen “just like that” (at least to my knowledge) –they are triggered by external energy influx from things like reactive oxygen species or ionising radiation. So yes, they cost energy but that energy doesn’t come energetically unstable chemical bonds; it is supplemented from external sources.

So it’s not necessarily true that mutations produce lower-energy chemical bonds.