It seems hopeless to try to provably find all the rational points on the genus $7$ curve $C$ mentioned in the question using either Chabauty's method, or etale descent. I will simply explain how to find a map from the genus $7$ curve to the rank three elliptic curve mentioned in the comment.
First, I noticed that all of the exponents on $a$ are even, and so setting $b = a^{2}$, we get an equation $b^{3} + b^2 (-18368 + 9184z - 2912z^{2}) + \cdots$. This equation defined a curve of $X$ genus $4$. I computed the automorphism group of this curve modulo several primes, and noticed that the order was $2$ (for $p > 17$ or so). I then had Magma compute the automorphism group $G$ over $\mathbb{Q}$ and found it had order $2$. If we realize $X \subseteq \mathbb{P}^{3}$ in its canonical model, the automorphism can be given by a simple linear change of variables. This allows one to easily compute the curve quotient, which results in a plane cubic. Magma's built-in routines can then get one to the minimal Weierstrass model of $X/G$.
Searching for nice equations for the maps at each step results in the map from the original curve (in terms of $a$ and $z$), to the homogenous form of the elliptic curve ($y^{2} z = x^{3} - 159600xz^{2} + 31482500z^{3}$) given by the equations
$$
x = a^{2} (470957175a^2 - 403726629600z^2 + 1240537082400z - 2481074164800)
$$
$$
y = 6385126825a^4 - 6377874641600a^2z^2 + 33875545316800a^2z -
67751090633600a^2 + 216129127244800z^4 - 8459911552153600z^3 +
99706100436096000z^2 - 331145109327155200z + 331145109327155200
$$
$$
z = a^{2} (1220184a^2 - 875686464z^2).
$$
If I hadn't tried several tricks to manually simplify the equations, they would have been several pages long.
