What gives us the certainty that the unit charge of a free electron - measured and confirmed with the highest precision in numerous experiments - is retained even when approaching and integrating into an atom and associated with the emission of EM radiation?

Subordinate questions, which should primarily serve to understand the above question: Has a measurement procedure been devised or already carried out that can experimentally investigate the charge or electric field strength of the bound electron? Would such research simplify the model of the fundamental electromagnetic interaction in the atom?