Per naturallyInconsistent's suggestion, I looked into some literature on hyperspherical harmonics. I could not find an explicit answer to this question however. But, I was able to get to what I believe is the right answer, which I will post here for posterity.
According to this post, for all $d>2$ the hyperspherical harmonics originating from harmonic polynomials of the same degree (i.e. with index $l_1$) form an irreducible representation of $SO(d)$, specifically the representation labeled by the Young tableau $(\ell_1,0,\dots,0)$ denoted $ U_{\ell_1}^d$. Now, since all Casimir invariants are multiples of the identity for irreducible representations, we can only identify $\ell_1$ from analyzing the $SO(d)$ Casimir invariants. We may as well only look at quadratic invariant, since it is easy to write and the eigenvalues are known.
To further classify the hyperspherical harmonics, an important observation is needed. First, in any construction of hyperspherical coordinates and therefore of hyperspherical harmonics, there is an implicit ordering of the variables, $x_1,x_2,\dots$ where an additional axial angle $\theta_i$ is needed to account for each successive $x_i$. The exceptions are the final two variables, which are both represented by a final azimuthal angle $\phi$. This ordering induces a chain of subgroups $SO(2)\subset \dots \subset SO(d-1)\subset SO(d)$. When restricted to $SO(d-1)$, $U_{\ell_1}^d$ becomes $\bigoplus_{\ell_2=0}^{\ell_1}U_{\ell_2}^{d-1}$. Each $U_{\ell_2}^{d-1}$ becomes $\bigoplus_{\ell_3=0}^{\ell_2}U_{\ell_3}^{d-2}$ when restricted to $SO(d-2)$ and so on and so forth. This continues until we get to the representations of $SO(2)$ which must be one dimensional and therefore the hyperspherical harmonics themselves. To find the $\ell_i$, we simply use the quadratic Casimir elements for each successive subgroup until $SO(3)$. For $SO(2)$, we use its angular momentum operator.
To translate that for those not well-versed in this math, here is the concrete answer to this question. First, identify the order of variables implicit in your definition of hyperspherical coordinates $x_1,x_2,\dots, x_{d-2},(x_{d-1},x_d)$. Then for each pair $1\leq i < j \leq d$, define the angular momentum operator $\hat{L}_{i,j} = (-1)^{j-i}(\hat{x}_i\hat{p}_j-\hat{p}_j\hat{x}_i)$.
The complete set of commuting observables consists of $\hat{L}_{d-1,d}$ along with the following operators labeled by $1\leq k \leq d-2$ $$\hat{\mathfrak{L}}_k=\sum_{k\leq i<j\leq d}\hat{L}_{i,j}^2$$
The eigenvalue relations for $|n,\ell_1,\ell_2,\dots,\ell_{d-1}\rangle$ are
$$\hat{\mathfrak{L}}_k|n,\ell_1,\ell_2,\dots,\ell_{d-1}\rangle=\hbar^2 \ell_{k}(\ell_k+d-k-1)|n,\ell_1,\ell_2,\dots,\ell_{d-1}\rangle$$
$$\hat{L}_{d-1,d}|n,\ell_1,\ell_2,\dots,\ell_{d-1}\rangle=\hbar \ell_{d-1}|n,\ell_1,\ell_2,\dots,\ell_{d-1}\rangle$$
