Case 1:
It is known that:
$$\sqrt[6]b\left(\frac{\text B_{\text I^{-1}_{\frac{3\sqrt[6]bx}{2\text{Im}(\omega_1)}-2}\left(\frac13,\frac12\right)}\left(\frac23,\frac12\right)}{6\sqrt[3]2}+\frac1{\text B}\right)=\zeta(x;0,-b) $$ with the Baxter B and $\omega_1$ constants along with the Weierstrass zeta $\zeta(x;a,b)$, incomplete beta $\text B_z(a,b)$ and inverse $\text I^{-1}_x(a,b)$ beta regularized $\text I_x(a,b$) functions. $\text I^{-1}_x\left(\frac13,\frac12\right)$ is expressible in terms of $\wp(x;a,b)$. Similarly, including inverse Weierstrass P $\wp^{-1}(x;a,b)$, there are conversion formulas $(1)$ for $\text B_x(a,b)$ into $\wp^{-1}(x;a,b)$.
However, using these formulas and this recurrence relation gives results for $a-n=\frac13,\frac14,\frac16,b=n+\frac12,n\in\Bbb N$, but not $a=\frac23$. Even using this parameter transformation $(2)$ cannot put $\text B_z\left(\frac23,\frac12\right)$ in terms of $\wp^{-1}(x;a,b)$. Therefore, the inverse of $\zeta(x;0,b)$ likely is inexpressible in terms of Weierstrass elliptic functions. Luckily, we have
$$\boxed{\zeta(y;0,b)=x\implies y=\frac{2\text{Im}(\omega_1)}{3\sqrt[6]{-b}}\left(\text I_{\text I^{-1}_{\frac{2\text Bx}{\sqrt[6]{-b}}-2}\left(\frac23, \frac12\right)}\left(\frac13,\frac12\right)+2\right)=\frac{2^{-\frac23}\text B_{\text I^{-1}_{2\left(\frac{\text Bx}{\sqrt[6]{-b}}\omega_1-1\right)}\left(\frac23,\frac12\right)}\left(\frac13,\frac12\right)+4\text{Im}(\omega_1)}{3\sqrt[6]{-b}},\frac{\sqrt[6]{-b}}{\text B}\le x\le \frac{3\sqrt[6]{-b}}{2\text B} }\tag3$$
Test it here. We could use the $\omega_2$ constant:
$$-\frac{(-1)^\frac23\text B_{-\frac{4z^3}b}\left(\frac13,\frac12\right)}{3\cdot2^\frac23\sqrt[6]b}-\frac{2\omega_2}{\sqrt 3\sqrt[6]b}=\wp^{-1}(z;0,-b) $$
Case 2:
appears with the complex conjugate $z$ and ubiquitous constant U:
$$\boxed{\zeta(y;a,0)=z\implies y=-\frac{\overline {\text B_\frac1{\text I^{-1}_{\frac{2iz}{\text U\sqrt[4]a}+i+1}\left(\frac34,\frac12\right)}\left(\frac14,\frac12\right)}}{2\sqrt2\sqrt[4]a},z=\frac{\text U\sqrt[4]a}2(i(1-x)-1),0\le x\le 1 }\tag 4$$
Shown here. Using $(2)$, $\text B_x\left(\frac14,\frac12\right)$ is expressible through Elliptic F and $\wp^{-1}(x;a,0)$ while $\text B_x\left(\frac34,\frac12\right)$ is through Elliptic E
Case 3:
$$\boxed{\zeta(y;0,b)=z\implies y=\frac{\sqrt[3]2\overline{\text B_\frac1{\text I^{-1}_{i\sqrt3-\frac{2\text B i z}{\sqrt[6]b}+1}\left(\frac23,\frac12\right)}\left(\frac16,\frac12\right)}}{6\sqrt[6]b},z=\frac{\sqrt[6]b((x-1)i+\sqrt3)}{2\text B},0\le x\le1}\tag5$$
Again, using $(2)$, $\text B_x\left(\frac16,\frac12\right)$ is expressible in terms of $\wp^{-1}(x;0,b)$
Question:
We can extend the domain a little:
$$\text I^{-1}_x\left(a,\frac12\right)=1-\left(2\text I^{-1}_\frac x2(a,a)-1\right)^2 $$
or simplify the $\text B_z(a,b)$ with $(2)$. Unfortunately, $(3),(4),(5)$’s domains are tight and inverting $\zeta(x;a,0)$ involves inverting Elliptic E, so no Weierstrass elliptic functions. How can one use symmetries and periodicity, etc. to invert, likely with $a$ or $b=0$, $\zeta(x;a,b)$ in closed form?
