I an assuming $a,b,c>0$. Let $\delta=\sqrt{b/a}$. Then $ \int_{-\infty}^\infty \frac{a\,b\,\operatorname{sinc}^2(c\,x)}{a+b\,\operatorname{sinc}^2(c\,x)}\,dx=\frac{b}{c}\int_{-\infty}^\infty\frac{\sin^2x}{x^2+\delta^2\sin^2x}\,dx=\frac{b}{c}\,I(\delta). $ Integrating the inequalities $ \frac{\sin^2x}{x^2+\delta^2}\le\frac{\sin^2x}{x^2+\delta^2\sin^2x}\le\frac{\sin^2x}{x^2} $ we get $ \frac{1-e^{-2\delta}}{2\,\delta}\,\pi\le I(\delta)\le\pi. $ This bounds are good only for for small $\delta$. For Large $\delta$ try the following: $ \begin{align*} I(\delta)&=2\int_0^\delta\frac{\sin^2x}{x^2+\delta^2\sin^2x}\,dx+2\int_\delta^\infty\frac{\sin^2x}{x^2+\delta^2\sin^2x}\,dx\\ &\le\frac{2\,\delta}{1+\delta^2}+2\int_\delta^\infty\frac{\sin^2x}{x^2}\,dx\\ &=\frac{2\,\delta }{\delta ^2+1}+\frac{1-\cos2\,\delta}{\delta}+{\pi -2\,\operatorname{Si}(2\,\delta )}, \end{align*} $ where $\operatorname{Si}$ is the sine integral. in the graph, $I(\delta)$ is in red and the upper and lower bound in blue.
