Main Question or Discussion Point
How come electrons don't lose mass as they radiate a field energy in the form of electromagnetic radiation???
I don't think that distinction is all that rigorous. I don't even think the distinction can be made which object "has" the energy...either the nucleon+electron individually or the atom as a whole...Potential energy is a property of a system (the atom), not of the individual particles (electron and proton) that make up the system.
I love the way you put that: that we know its effects but not what causes it.To the OP:
In a large sense you are attempting an understanding of "mass"
That's totally fine, but be aware that all we currently know about mass is it's effects.
We DO NOT KNOW what causes it.
Hopefully, LHC experiments can assist in our understanding.