If $g_j$ is SOS, i.e. $g_j=\sum_k h_k^2$, then $g_j(x)\leq 0$ iff $g_j(x)=0$, i.e. $h_1(x)=h_2(x)=\dots =0$. So this is a general case, although with equality constraints only.

A more interesting question would be to allow inhomogenous constraints $g_j(x)\leq b_j$, with $b_j\in\mathbb{R}$. IIRC, this would be solvable with just one step of the Lassere's hierarchy, as the whole point of it is approximating arbitrary nonnegative polynomials by SOS.
(I believe the latest Lasserre's [book](https://doi.org/10.1017/CBO9781107447226) is discussing this in Chapter 11).