The product of the periods associated to $f$ ($\omega^{\pm}$ in your notation) is equal modulo $\mathbf{Q}(f)^\times$ to $i \langle f, f\rangle$ times some explicit powers of $\pi$ depending on the weight. So the result follows from the rationality theorems of Manin (and Shimura), exactly as you suggest.

For an eigenform, the product of the periods associated to $f$ ($\omega^{\pm}$ in your notation) is equal modulo $\mathbf{Q}(f)^\times$ to $i^{1-k} G(\chi) \langle f, f\rangle$, where $G(\chi)$ is the Gauss sum of the nebentypus character of $f$, and $k$ is the weight. This is Theorem 1 (iv) of Shimura's On the periods of modular forms (Math. Ann. 229, 1977).

Since the periods of the twist $f_\varepsilon$ are $G(\varepsilon)$ times the periods of $f$, and $G(\varepsilon^2 \chi) / [G(\chi) G(\varepsilon)^2]$ lies in the field of values of $\chi$ and $\varepsilon$, it follows that the ratio $\langle f_\varepsilon, f_\varepsilon\rangle / \langle f, f \rangle$ lies in $\mathbf{Q}(f, \varepsilon)^\times$ (and hence in $\mathbf{Q}(f)^\times$ if $\varepsilon$ is quadratic).

(Are you sure the statement is true for non-eigenforms? It seems very unlikely, somehow.)