Given that $\{a_{n}\}$ is a geometric sequence, we can use the properties of geometric sequences to find relationships between its terms. Specifically, for a geometric sequence, the product of terms equidistant from the ends is constant. This means that $a_{2}a_{6}$ and $a_{3}a_{5}$ should be equal to ${a}_{4}^{2}$, as they are equidistant pairs.

Starting from the given equation:
$$a_{2}a_{6} + 2{a_{4}}^{2} = \pi$$

Given the property of geometric sequences:
$$a_{2}a_{6} = a_{3}a_{5} = {a}_{4}^{2}$$

Substituting $a_{3}a_{5} = {a}_{4}^{2}$ into the equation, we get:
$$a_{2}a_{6} + 2{a_{4}}^{2} = 3a_{3}a_{5} = \pi$$

From this, we can solve for $a_{3}a_{5}$:
$$a_{3}a_{5} = \frac{\pi}{3}$$

Knowing that $\tan\left(\frac{\pi}{3}\right) = \sqrt{3}$, we can find the value of $\tan(a_{3}a_{5})$:
$$\tan(a_{3}a_{5}) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3}$$

Therefore, the correct answer is:
$$\boxed{A}$$