Need to calculate $\frac{E\left[|x(n)|^4\right]}{E\left[x^2(n)\right]}$
The most efficient way that I can think of to do it would be as follows:


For each input sample $x[n]$, calculate $|x[n]|^2$. This is more efficient than calculating $|x[n]|$, because you don't have to do the square root:


$$
|x[n]| = \sqrt{\text{Re}\{x[n]\}^2+\text{Im}\{x[n]\}^2} \\
|x[n]|^2 = \text{Re}\{x[n]\}^2+\text{Im}\{x[n]\}^2
$$


Average the instantaneous power estimate $|x[n]|^2$ as needed for your application (you mentioned exponential averaging).
Note that you can also get $|x[n]|^4$ efficiently using this method; you just square the measurement of $|x[n]|^2$ that you made above, then average again as desired.