Is there a simpler way to calculate the amplitude response of the following filter
The given transfer function has a specific form. With the denominator polynomial $A(z)=2+bz^{-2}$, you can write the transfer function as

$$H(z)=\frac{z^{-2}}{2}\frac{A\left(\frac{1}{z}\right)}{A(z)}\tag{1}$$

Since for $z=e^{j\omega}$ (i.e., on the unit circle, where we evaluate the frequency response), we have $|A(z)|=|A(z^{-1})|$, the magnitude of $H(e^{j\omega})$ is given by

$$\left|H(e^{j\omega})\right|=\frac12\tag{2}$$

The given filter is a (scaled) all-pass filter. It's easy to recognize all-pass filters by noting that their numerator and denominator polynomials have mirrored coefficients.