> estimating how far every ... baryon ... is from all the others

The true metric of the universe, sourced by every massive object (and every massless one, and all self-energies and interactions and fluxes of momentum-energy), is fantastically complicated.

The standard cosmology is tractable only because it coarse-grains all this into a model where at every point in space (having taken a particular slicing of the whole spacetime into spatial volumes ("space") indexed by the scale factor) has a total energy-density. The total is a sum that includes the energy-density from baryons, radiation, dark matter, and so forth. The energy-density in any given slice is modelled as the same at every point in the spatial slice, leading to the observed large-scale isotropy and homogeneity that is central to the model, and allows for a cosmological frame to be picked out.

In the cosmological frame, where coordinates expand with the metric expansion of space, we can talk about the energy-density at a given point in space. In principle we can make measurements at a large number of points and produce an average energy-density. Finally, we can talk about the consequences of the average value: https://www.astronomy.swin.edu.au/cosmos/C/Critical+Density

So, in a sense, cosmologists do talk about whether the entire universe will recollapse or expand forever (or fall into some steady state), and (average) baryon-density is an important factor. Thus there is some upper bound for a local quantity not a million miles from gravitational potential energy. (Below your question there are others pointing out that there is no such global quantity available. This mainly means that while one can imagine increasing the average baryon or DM density leading to a collapse of the whole universe, this is still firmly in the FLRW universe, and definitely not converting from FLRW to an asymptotically flat spacetime around a central collapsing mass. But see below.)

Going further, we can evolve the (average) baryon-density in a space along the scale factor. In the timelike (scale factor) direction away from baryogenesis, the energy-density of baryons decreases. At a finer-grained level that means clouds of neutral atoms and molecules tend to thin out. (So does radiation, so does dark matter, so do relic neutrinos, and so forth; pretty much everything but the cosmological constant (which is constant, after all) falls to zero on average far enough from the formation of the cosmic microwave background).

This coarse-grained picture can be refined in several ways, by e.g. introducing inhomogeneities: overdensities or underdensities of baryons, for example, which evolve into large black holes and voids respectively. One can then ask questions in an inhomogeneous cosmology about the (average) density of black holes of various masses, and compare that to the energy-densities of baryons and the rest. The question is, for a given spatial slice, what fraction of baryons have fallen into black holes, and what fraction has not? Then evolve that question along the scale factor (e.g., in the future, are most baryons in black holes, or are they mostly spread out in wispy filaments around the edges of large voids?).

One can do some headstands and try to understand the fraction of all baryons not yet fallen into black holes in that sort of cosmology as relating to gravitational potential energy, however I think that it's bound to be more useful to think, like above, of the (averaged) energy-densities and how that averaged (and thus coarse-grained) picture generates a metric comparable to FLRW. Instead, interpreting your question somewhat, you might start with something that ultimately must be nonuniformities in a contraction of the Riemann curvature tensor (see <https://en.wikipedia.org/wiki/Scalar_curvature>) then dusting that geometry with objects whose trajectories you'd study so that in some set of coordinates you could carve out (for each of them) from their intrinsic mass a kinetic energy and potential energy.

I don't think that approach would be fundamentally wrong. It is essentially along the lines of Lagrangian mechanics (L = T - V, where V is a potential energy), although in a general-relativistic setting this gets hard, see <https://en.wikipedia.org/wiki/Relativistic_Lagrangian_mechan...>. Inevitably you would have to do some coarse-graining for tractability, and would want to make sure your coarse-graining procedure is not unphysical.

Finally, apologies for not expanding a number of acronyms, and for wandering off-track a bit. Yours was an interesting question especially in light of some of the more technical replies below, and I was torn about what audience-expertise to write for (and settled on probably satisfying nobody).