Hierarchy of electronic wavefunctions
I've been reading up on this, and will take a shot at the answer. It is going to be an interesting answer, and probably not what you expect. There has been an interesting conversation going on in the Journal of Chemical Education in response to this letter advocating the retirement of the localized hybrid orbitals because they "aren't real" and cannot be used to predict "all properties" of molecules. Rebuttals included comments that delocalized molecular orbitals also fail to predict "all properties". However, the final word in the discussion came from this letter, which reminds us what orbitals are.

Orbitals are not real. The electrons that populate orbitals are real. Orbitals are mathematical constructs created to describe the curious wave-like probabilistic quantum mechanical properties of electrons. When we talk about atomic orbitals, we generally mean the 1-electron hydrogenic atomic orbitals (HAOs), which are mathematically represented as a wavefunction ($\Psi$) and graphically represented by the probability density function ($\Psi ^ * \Psi$) at either 90% or 95% level. Despite valiant and massive efforts, wavefunctions for multi-electron atoms cannot be solved exactly. They are approximated as a parameterized HAOs. When we talk about the atomic orbitals of carbon, we usually mean the HAOs that closely approximate what we think the AOs of carbon should be. 

Molecules and their orbitals are even more troublesome. The LCAO approach creates a bunch of nice 1-electron wavefunctions out of the set of HAOs. These methods give rise to both the canonical delocalized molecules orbitals (CMOs), and the localized valence bond orbitals (VBOs). As the authors of that letter point out:


  The term canonical implies that CMOs are not the only possible MOs, and there are many orbitals that generate an identical “chemical reality”.


The approximate polyelectron wavefunction of a molecule (at least in HF theory) is the Slater determinant of a series of 1-electron CMOs $\psi _i$:
$$  \psi _1 \overline{\psi} _1 \psi _2 \overline{\psi} _2 \psi _3 \overline{\psi} _3 \psi _4 \overline{\psi} _4 ... $$

An interesting property of Slater determinants is that unitary operations (like rotations) on the set of CMOs return a different set of MOs, which produce the same expectation values for the polyelectron wave function as original set:

$$ \psi _1 \overline{\psi} _1 \psi _2 \overline{\psi} _2 \psi _3 \overline{\psi} _3 \psi _4 \overline{\psi} _4 ...~~ \underrightarrow{unitary~~transformation} ~~\psi _a \overline{\psi} _a \psi _b \overline{\psi} _b \psi _c \overline{\psi} _c \psi _d \overline{\psi} _d ...$$

Mathematically, it seems to me, that a linear combination of MOs that are themselves linear combinations of AOs would still be a linear combination of AOs, just a different one than you started with. If that is true, then all you have is a different set of MOs than you started with, and according to HF theory, that should give you the same expectation values. 

Thus a set of "LCMOs" are no more or less valid than the original set of LCAOs. They likely do not produce any additional insights into the structure of the molecule. However, the approach might be computationally more efficient when trying to determine the MOs of very large molecules and polymers.