= Solution
An explicit answer, initially written where $yz\ne0$, is the <meromorphic differential on a Riemann surface>
$$
\boxed{\eta=\frac{d(x/z)}{y/z}=\frac{z\,dx-x\,dz}{yz}.}
$$
The ratios $x/z$ and $y/z$ are functions on $C$, so the expression does not depend on the representative of its <homogeneous coordinates>. It is a scalar multiple of an <invariant differential on an elliptic curve>.
On the <complex torus> $\mathbb C/\Lambda$, the differential $dw$ descends from $\mathbb C$ because translation by any element of the <period lattice> leaves $dw$ unchanged. It is <holomorphic> and nowhere zero. The given <biholomorphism> satisfies, wherever the displayed quotient is initially defined,
$$
\Phi^*\eta=\frac{d\wp(w)}{\wp'(w)}=dw.
$$
Consequently, $\eta=(\Phi^{-1})^*(dw)$ extends across every apparent denominator zero and across $[0:1:0]$ as a nonzero <holomorphic differential form> on all of $C$.
One can also check the apparent exceptional points directly. On $z=1$, put $P(X)=4\prod_i(X-e_i)$, so $Y^2=P(X)$. At $Y=0$ the roots are distinct, and differentiating gives
$$
\frac{dX}{Y}=\frac{2\,dY}{P'(X)},
$$
which is regular and nonvanishing there. At $[0:1:0]$, use $u=x/y$, $v=z/y$. The equation is $v=4\prod_i(u-e_iv)$, so $u$ is a <local parameter on a smooth algebraic curve> and $v=4u^3+O(u^5)$. Hence
$$
\eta=v\,d(u/v)=du-\frac{u}{v}\,dv=(-2+O(u^2))\,du,
$$
also regular and nonvanishing.
Only distinctness of the $e_i$ is needed. The paper additionally writes $e_i\ne0$, which is not true for every <period lattice>: for $\Lambda=\mathbb Z+\mathrm i\mathbb Z$, the symmetry $\wp(\mathrm iw)=-\wp(w)$ makes the half-period value at $w=(1+\mathrm i)/2$ zero, since $\mathrm iw\equiv w\pmod\Lambda$. This unnecessary restriction has no effect on the answer.
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