Conservation of Energy Down The Drain?

You can't. At first, I thought that's what you were doing - but on closer inspection, you seemed to be offering a definition of a number, which I understood to be as follows:

Divide space-time up into a bunch of small pieces. In each small piece:

Take the energy density rho in some local frame
multiply it by the volume element in said local frame

Sum them all together .....

The safest remark is probably 2). However, Noether's theorem, which was created to address this issue, suggests that if we did have a way of defining energy on a global scale, we'd have as a result some "preferred" frame. (At least, that's my understanding). We'd have a preferred four-parameter subgroup of space-time.

Preferred frame theories are possible, but are out of vogue - and are not GR. For instance, Self Creation Cosmology would be an example of a non-GR theory that would have a conserved energy, getting around Noether's theorem by having a scalar field with an associated Jordan Frame, a bit like Branse-Dicke theory.

Unfortunately, both Branse-Dicke theories and Self-Creation Cosmology (as originally proposed) appear to be inconsistent with experiment - the later predicted different results for Gravity probe B.

So, unless my understanding is wrong, GR won't have a conserved energy because it doesn't have a preferred frame. Examples of theories that have a conserved energy also have preferred frames.