Any theory that extends general relativity must reduce to GR in some limit, and must (of course) agree with experiment. Probably the simplest such extension is Brans-Dicke gravity, which corresponds to replacing the gravitational constant by a scalar field on spacetime: $\frac1G\to\phi(x)$. The kinetic term for $\phi$ in the Lagrangian, appropriately dimensionalised, has a prefactor $\omega$. This field now couples to the Ricci scalar in the Lagrangian, and so has an equation of motion which is determined by the geometry and hence the energy-momentum content. Note that the only free parameter in the theory is $\omega$, which must be constrained by experimental bounds. Current observational evidence suggests $\omega>40\,000$ - for comparison, general relativity is heuristically reobtained in the $\omega\to\infty$ limit. Here's the equation of motion for $\phi(x)$:
$$ \square\phi=\frac{8\pi}{2\omega+3}T_\alpha^\alpha $$
which is a wave equation for massless $\phi$, sourced by the trace of the energy-momentum tensor. This means that $\phi$-waves propagate at the speed of light and, asymptotically, $\square\phi=O(\rho/\omega)$ for dust with matter density $\rho$. Now in the new "Einstein equations" relating $R_{\mu\nu}$ and $T_{\mu\nu}$, you can interpret $\phi$ and its variations either as a "generalised mass" or a "generalised spacetime curvature" - either way, in the Newtonian limit, it is manifest that this is equivalent to a changing gravitational coupling (even though the Einstein equations themselves have become rather more complicated).
With the reasonable additional assumption that $\phi$ approaches a constant value $\tilde\phi$ asymptotically, we can expand around this value in powers of $\frac{1}{\omega}$. This means that $\phi\sim\tilde\phi+O(1/\omega)$ (provided that the trace of the energy-momentum tensor does not vanish identically, see [1]). Since, as mentioned, $\omega>40\,000$, we would expect only very small variations in the gravitational strength. However, due to their long-range nature and luminal travel, it is plausible that many such waves, generated from various sources and different spacetime locations superpose to reduce $G$ in a region (bearing in mind that the paths of these fluctuations are themselves affected by the spacetime geometry, and so lens naturally into a dense region), or that there is some (possibly coincidental) fine-tuning. A good review is given in [2], albeit in asymptotically de-Sitter space (although a similar analysis applies).
This certainly isn't going to resolve the singularity problem (indeed, if any theory that agrees with experiment could, you would see a much larger uptake of it), but it does provide a mechanism to reduce the gravitational strength in regions of high density. Pertinent to your cosmological case is the possibility that $\omega$ itself is not a constant, and has stabilised at a high value over time - thus allowing $\phi$-waves of greater magnitude, at that epoch creating regions of highly varying gravitational strength.
References:
- [1] Banerjee, Sen (1997)
- [2] Ozër, Delice, Gravitational waves in Brans-Dicke Theory with a cosmological constant