You have it backwards. The lowest entropy state of the universe would be if there were no attractive forces, only repellent forces, as then all particles would be forced into something of an expanding lattice, but with all particles equidistant from all nearest neighbors (of the same type).
It is gravity which disrupts this and causes clumping, and that _increases_ entropy.
I know it's confusing because normally one would think of a cloud of gas as more disordered than the star it might collapse into, but that is not so. For one the star would be much hotter, and the motions of every particle in the star much more chaotic.
If everything must be constrained to the lattice points, yes. However, empty space has high Boltzmann entropy: you can cut a patch of empty space from here and swap it for the same volume of empty space from there, and the two coarse grain macrostates will be indistinguishable.
Expanding de Sitter quasi-vacuum has tremendous growth in entropy. Gibbons and Hawking gives this (for 3+1d de Sitter) as a quarter of the horizon area: S_H = \frac{Area_{H}}{4} \sim H^{-2} with the "quasi-" giving us increasing growth in the horizon area as DoFs exit the horizon compared to classical pure de Sitter vacuum.
I'm not sure how confining some species of matter to expanding lattice is different from quasi-vacuum in the limit where the lattice spacing is large. I guess you have to abolish continuum spacetime in favour of a taxicab geometry with an analogue of dark energy? Otherwise, how does it differ from an isotropic homogeneous FLRW dust?
Oh, and by the way, entropy is the evolution of a system given the forces in it. So yes, in a universe with only repellent forces at first it would have low entropy (like ours did) and then as the particles get forced into an ever emptier lattice the universe would have more entropy. It's the forces -attractive or repellent- that make the system evolutions possible that lead to higher entropy over time.
If you consider a universe with only protons in it initially clearly they would all be forced into a low-entropy lattice and space would expand. Eventually the space between them would grow to be sufficiently large and empty that its own entropy would be enormous. But that doesn't deny that gravitational collapse is an entropy-increasing mechanism.