There seems to be a problem with Klein's septic equation $(2)$ combined with the purported roots in $(3)$. Let $\,k\,$ be any integer. Define $$ P_1(k) := \gamma^{k}\mu^2 + \gamma^{2k}\lambda^2 + \gamma^{4k}\nu^2 \tag{1} $$ and $$ P_2(k) := \gamma^{3k}\lambda\nu + \gamma^{6k}\mu\nu + \gamma^{5k}\lambda\mu. \tag{2} $$ Let $$ \,a := 2\sqrt{-7}/\eta(\tau)^2 \; \textrm{ and } \; b := -(7+\sqrt{-7})/\eta(\tau)^2. \tag{3} $$ Define the roots of $(2)$ as $$ r_k := a\,P_1(k) + b\,P_2(k). \tag{4} $$ The polynomial $\, P(z) := (z - r_1)(z - r_2)\cdots (z - r_7)\,$ expands to $$ P(z) = z^7 \!-\! 2^2\! \cdot\! 7^2\, (7+\sqrt{-7})\, z^4 \!+\! 2^5\!\cdot \!7^4\, (5+\sqrt{-7})\,z \!+\! 2^7\! \cdot\! 7^3 \sqrt{-7} \frac{g_2(\tau)}{\eta(\tau)^8}. \tag{5}$$ This was for values of $\,\lambda,\mu,\nu\,$ as suggested by Tito Piezas III.
P.S. Note that $\, \sqrt[3]{\Delta} = \eta(\tau)^8$ is the denominator of the constant term of $P(z).$ Also note that $\, 1728 J(\tau) = j(\tau) = g_2(\tau)^3/\eta(q)^{24}\,$ where $\,J(\tau)\,$ is Klein's invariant. Thus using cube roots $\, 12\sqrt[3]{J(\tau)} = g_2(\tau)/\eta(\tau)^8\,$ and the constant term can be written as $\, 2^9\!\cdot 3 \cdot\! 7^3 \sqrt{-7} \sqrt[3]{J(\tau)} \,$ which is closer to Klein's version.
P.P.S. I see that Tito has used a set of roots $\,y_k\,$ differing by a common factor from $\,z_k\,$ to simplify the septic. Perhaps Klein would have used that version, but he preferred $\,J(\tau)\,$ instead of $\,j(\tau).$