The density of primes dividing a particular non-linear recurrence sequence

Define the sequence $\{b_n\}$ by $b_0=1,b_1=2, b_2=1,b_3=-3$, and $$b_n=\begin{cases} \frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\not\equiv 2\pmod 3,\\ \frac{b_{n-1}b_{n-3}-3b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\equiv 2\pmod 3. \end{cases}$$ We relate this sequence $\{b_n\}$ to the coordinates of points on the elliptic curve $E:y^2+y=x^3-3x+4$. We use Galois representations attached to $E$ to prove that the density of primes dividing a term in this sequence is equal to $\frac{179}{336}$. Furthermore, we describe an infinite family of elliptic curves whose Galois images match that of $E$.