Remember that $S^3=SO(4)/SO(3)=:G/K$ as a symmetric space. The following actually holds for every compact symmetric space $M=G/K$.
Take the Cartan decomposition $\mathfrak g=\mathfrak k\oplus \mathfrak p$. By using that the vector bundle $\mathcal S_0^2(M)$ of traceless symmetric 2-forms is homogeneous (i.e. $\mathcal S_0^2(M)\simeq G\times_{\tau} Sym^2(\mathcal m^{\mathbb C})$, where $\tau$ denotes the $K$-action on $Sym^2(\mathcal m^{\mathbb C})$), it decomposes as $$ \mathcal S_0^2(M)^{\mathbb C} = \widehat{\bigoplus}_{\pi\in\widehat {G}} V_\pi\otimes Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C}). $$ Of course, this sum is reduced to those $\pi\in\widehat G$ such that $Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C})\neq0$.
Now, since you are interested on $\Gamma\backslash S^3 = \Gamma\backslash G/K$ with $\Gamma$ a discrete subgroup of $G$, I guess that $$ \mathcal S_0^2(\Gamma\backslash M)^{\mathbb C} = \widehat{\bigoplus}_{\pi\in\widehat {G}} V_\pi^{\Gamma}\otimes Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C}). $$ In conclusion, the sum is now restricted to those $\pi\in\widehat G$ such that $Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C})\neq0$ and $V_\pi^{\Gamma}\neq 0$.
The Lichnerowicz Laplacian on $\mathcal S_0^2(M)$ (or $\mathcal S_0^2(\Gamma\backslash M)$) acts by the casimir element (here, it is crucial that $G/K$ is symmetric), so each term $V_\pi^{\Gamma}\otimes Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C})$ contributes to the spectrum of the Lichnerowicz Laplacian with the eigenvalue $\lambda_\pi$ such that $Cas\cdot V_\pi = \lambda_\pi Id_{V_\pi}$ (which can be computed with Freudenthal's formula) with multiplicity $\dim V_\pi^{\Gamma}\dim Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C})$.
One has $$ \lambda_1(G/K) =\min_\pi \lambda_\pi, $$ where $\pi$ runs over $\widehat G$ such that $\dim Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C})>0$, and at least one element in $\mathcal S_0^2(M)$ induced by $V_\pi\otimes Hom_K(V_\pi,Sym_0^2(\mathfrak p)^{\mathbb C})$ is divergence-free (note that I haven't mentioned this condition before).
Concerning your question about whether $\lambda_1(\Gamma\backslash G/K)=\lambda_1(G/K)$ for a given $\Gamma\subset G$, it reduces to check whether $\dim V_\pi^\Gamma>0$ for any $\pi$ attaining the displayed identity above, that is, $\lambda_1(G/K)=\lambda_\pi$.
For the sphere $G/K=S^3$, I think that $\lambda_1(S^3)=12$ (rather than 6 as you claimed, see for instance Example 3.1.2 in Klaus Kröncke's PhD thesis), and it is attainded only by $\pi_{2\varepsilon_1}$ (the irrep of $SO(4)$ having highest weight $2\varepsilon_1$), which is equivalent to $Sym_0^2(\mathbb C^4)$. In conclusion, $\lambda_1(\Gamma\backslash S^3)=12$ if and only if $\dim V_\pi^{\Gamma}=\dim Sym_0^2(\mathbb C^4)^{\Gamma}>0$.