Does this random variable have a density?
Yes $Z$ has a density. For each $s$ the random variable $B_s^4$ is a linear combination of Hermite polynomials of degree at most $4$. Therefore the random variable $$ I_n = \frac{1}{2^n} \sum_{i=1}^{2^n} B_{i/2^n}^4 - B_{(i-1)/2^n}^4 $$ is also is a finite sum of Wiener chaos. Now $I_{n} \to Z$ in $L^2$ therefore $Z$ is also in a finite sum of chaos. You can then you can use a result of Shigekawa to conclude.
But I would not be surprise if a simpler argument exists.