The following construction reduces your problem it to a classical and well-studied problem in invariant theory. First, I claim that there is a natural way to interpret an $n$-tuple of points in $S^2$ as a point of $\mathbb{CP}^n$ and vice-versa. This depends on interpreting $S^2$ as $\mathbb{CP}^1$ and the classical fact that the symmetric product of $n$ copies of $\mathbb{CP}^1$ is naturally regarded as $\mathbb{CP}^n$. Moreover, this can be done in an $\mathrm{SO}(3)$-equivariant way, so that your problem becomes that of finding invariants for an action of $\mathrm{SO}(3)$ on $\mathbb{CP}^n$, a very classical problem.
To see this, remember that $\mathrm{SU}(2)$, which is the double cover of $\mathrm{SO}(3)$ acts on $\mathbb{C}^2$ in the obvious way and that this action is transitive on the $1$-dimensional subspaces $L\subset\mathbb{C}^2$, the set of which is naturally $S^2=\mathbb{CP}^1$. Moreover, the induced action on $S^2$ is that of $\mathrm{SO(3)}=\mathrm{SU}(2)/\{\pm I_2\}$. Thus, nonzero vectors in $\mathbb{C}^2$ up to nonzero complex multiples correspond to points of $S^2$, and, given any $n$-tuple of unit vectors $u_i\in S^2$ for $1\le i\le n$, they can be represented by an $n$-tuple of lines in $\mathbb{C}^2$ of the form $u_i = [v_i] = \mathbb{C}\cdot v_i\subset \mathbb{C}^2$. Now consider the symmetric product
$$
v_1v_2\cdots v_n\in \mathsf{S}^n(\mathbb{C}^2) \simeq \mathbb{C}^{n+1}
$$
The line spanned by this product, $[v_1v_2\cdots v_n]\in \mathbb{CP}^n$ is well-defined, independent of the choice of the $v_i$ to represent the $u_i$. Conversely, since any nonzero complex polynomial in two variables that is homogeneous of degree $n$ can be factored into linear factors uniquely up to complex multiples and permutations, it follows that any element $p\in \mathbb{CP}^n$ can be constructed this way and, moreover, the $n$-tuple of points $u_i\in S^2 = \mathbb{CP}^1$ that gives rise to $p$ is uniquely determined up to permutation.
Now, what about the action of $\mathrm{SO}(3)$? Since this is induced by the action of $\mathrm{SU}(2)$ on $\mathbb{C}^2$, and that action extends in the usual way to the symmetric power $\mathsf{S}^2(\mathbb{C}^2)=\mathbb{C}^{n+1}$, on which it is irreducible (as a complex representation), it follows that this construction is $\mathrm{SO}(3)$-equivariant.
Thus, you are reduced to finding a 'complete' (in your sense) and 'independent' (in your sense) set of invariants for the action of $\mathrm{SO}(3)$ on $\mathbb{CP}^n$ that is induced by the irreducible representation of $\mathrm{SU}(2)$ on $\mathbb{C}^{n+1}$. This is, of course, a very classical problem, about which an enormous amount has been known since the 19th century. In particular, the Clebsch-Gordan formulae can be used to give one a procedure for generating all of the polynomial invariants, but they inevitably have complicated relations among them as soon as $n$ gets bigger than $3$ or $4$.
