There are other gauge theoretic obstructions to the existence of rational homology cobordisms. See eg Furuta (Invent. Math., 100 (1990), 339–355 Fintushel-Stern (Topology, 26 (1987), 385–393), Matic (J. Differential Geom. 28 (1988), no. 2, 277–307) and my paper (Topology 27 (1988), no. 4, 401–414.) The two latter papers show that certain invariants $\rho_\alpha$ of the boundary $3$-manifolds associated to homomorphisms $\alpha$ to $U(1)$ must vanish for a rational homology cobordism. I believe that these kinds of obstructions are independent from the $d$ and $h$-invariants.

One thing to keep in mind in discussing the $d$ and $h$-invariants is that they are not really invariants of rational homology cobordism, but rather of $spin^c$ rational homology cobordism. A similar point holds for obstructions arising from $\rho_\alpha$ invariants, because to compare invariants of boundary components of a rational homology cobordism, you need that the representation on $\pi_1$ of the boundary extend over the interior. This is more or less the same point as requiring that some $spin^c$ structure on the boundary extends over the interior.

If you are willing to consider spin rational homology cobordism, then of course there's the Rochlin invariant.

Finally, the phrasing of the question suggests that you are interested in *smooth* cobordisms. I don't know any obstructions to *topological* rational homology cobordism apart from some simple things about linking forms.