Considering that on the atomic level objects consists of densely spaced positively and negatively charged particles, does not the acceleration of those objects lead to Bremsstrahlung of those particles? And although the monopole field is zero, couldn't higher order multipole radiation escape and cause the inertia? I would think at least a small effect like this must happen, even if it doesn't explain all of the inertia.
No, Bremsstrahlung does not cause inertia.
The power radiated by Bremsstrahlung scales as charge but not mass. Also, it scales as acceleration squared.
This does not reduce to the classical inertia $m\vec a$.