Also consider $f= 1_{[-1/2,1/2]}$. Then $\widehat{f}(t) = \mathrm{sinc}(t)$, and so
$$\langle \eta\ast f, f\rangle = \langle \widehat{\eta\ast f},\widehat{f}\rangle = \int_{-\infty}^\infty e^{-\epsilon |t|} \mathrm{sinc}(t)^2 dt.$$
Here the approximation is trickier because of convergence issues; I'll finish this later. I think there's a closed expression that the symbolic integrators I have barely fail to find. EDIT: silly me, of course one wants to work in physical space for that $f$:
$$\begin{aligned}\langle \eta, f\ast f\rangle &= 2 \int_0^\infty \eta(t) dt - 2 \int_0^1 \eta(t) t dt = 1 - 4\epsilon\cdot \int_0^1 \frac{t dt}{\epsilon^2 + (2\pi t)^2}\\ &= 1 - \frac{\epsilon}{2\pi^2}\cdot \log \left(1 + \left(\frac{2\pi}{\epsilon}\right)^2\right),\end{aligned}$$ and so $$I_\epsilon(f) = \frac{\epsilon}{2\pi^2}\cdot \log \left(1 + \left(\frac{2\pi}{\epsilon}\right)^2\right),$$ which, for very small $\epsilon>0$, is much worse than the $\cos$ example above; it is not even $O(\epsilon)$.
