Since you allow virtual characters you should definitely expect such a thing (due to the general philosophy of writing down Eisenstein series as linear combinations of theta series after Siegel, Weil and others).
Here is an explicit construction. Take the simplest affine Kac-Moody Lie algebra, namely $A_1^{(1)}$, and take the level to be $1$. Then there are (essentially) two integrable highest weight $A_1^{(1)}$-modules of this level. Let's denote them by $V_1$ and $V_2$ for simplicity.
As usual,let $$\theta_{00}(q)=\sum_{n \in \mathbb{Z}} q^{n^2}, \ \ \theta_{10}(q)=\sum_{n \in \mathbb{Z}} q^{(n+1/2)^2}.$$ Then the corresponding (homogeneous) characters are $\chi(V_1)(q)=\frac{\theta_{00}}{\eta}$ and $\chi(V_2)(q)=\frac{\theta_{10}}{\eta}$.
You can easily show that
$$E_6=-33 \theta_{00}^4 \theta_{10}^8+ \theta_{00}^{12}+\theta_{10}^{12}-33 \theta_{00}^8 \theta_{10}^4$$
Now $\frac{E_6}{\eta^{12}}$ is just a linear combination of level $12$ integrable $A_1^{(1)}$-modules (view each summand as a tensor product of 12 level one modules).If you believe in "level-rank duality" similar construction should be possible for $A_{12}^{(1)}$ at level one, I think.
One more thing. Your quotient reminds me of Serre's paper "Sur la lacunarite des puissances de $\eta$", on the lacunarity of even powers of the $\eta$-function. In the case of $\eta^{14}$, he uses a nice identity
$$\frac{E_6}{\eta^{12}}=\frac{\varphi_{K,c_+}+\varphi_{K,c_-}}{\eta^{14}},$$ where $\varphi_{K,c_{\pm}}$ are certain CM modular forms of weight $7$ (the field is $K=\mathbb{Q}(\sqrt{-3})$ and $c_\pm$ are Hecke characters). I wonder if the right-hand side can be linked to anything in representation theory.
