Also, since the field is not static, but moves at c, differences in energy densities must be perpetuated by the body. According to your theory, how is the energy density perpetually lessened with proximity to the body?[\QUOTE]
For a static source the field is static. This theory aims to be an alternative continuous limit for a future quantum theory of gravity.
When that theory arises, gravitons will be radiated out, and an explanation for their regeneration will then be needed. Unless gravitons are like virtual-photons that exist, but don't really exist because the energy does not add up, but our math predicts exactly the right results when we use them.
Finally, you examine the energy density of the field.
The binding energy is not calculated by integrating potential energy over a volume. One must integrate the work done to produce that potential to start with.
The correct form for the energy density of the field is given in the OP.
A derivation of this can be found in a web search for the energy density of an electric field.