Numerology with Ramanujan's pi formula
(Too long for a comment.) After staring hard at my question and recalling an old MSE post of mine, I made an inspired guess and found,
Level 8
$$\frac{1}{\pi}=\frac{192\sqrt{2}}{(396^2+4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k \color{blue}\alpha^{k-2j} \tbinom{k}{2j}\tbinom{2j}{j}\tbinom{2j}{j} \frac{58\cdot15015k+72798-37\color{blue}\alpha/4}{(396^2+4\color{blue}\alpha)^k}$$
for general real $\alpha$, so turns out it is unnecessary to restrict it to powers of $2$. (Thus, $s_5(k)$ and $s_6(k)$ do have a closed-form and are integer sequences.)
Level 9
Similarly, the Chudnovsky formula (a level 1 Ramanujan-Sato) yields,
$$\frac{1}{\pi}=\frac{12}{(640320-4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{3j}\tbinom{2j}{j}\tbinom{3j}{j} \frac{163A\,k+B+1448\color{blue}\alpha/3}{(-640320+4\color{blue}\alpha)^k}$$
where $A=1114806, B=13591409$.
(Added later.) Now that I know what to look for, one can find other families.
Level 6
For example, starting with H. H. Chan, W. Zudilin, et al's
$$\frac1{\pi}=\frac{192\sqrt3}{(2\cdot140^2)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}k\sum_{j=0}^k\tbinom{k}{j}^3\,\frac{140(561k+53)}{(2\cdot140^2)^k}$$
mentioned in H. H. Chan and S. Cooper's "Rational analogues of Ramanujan's series for 1/π", we find, $$\frac1{\pi}=\frac{192\sqrt3}{(2\cdot140^2+4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}k\sum_{j=0}^k \color{blue}\alpha^{k-j}\tbinom{k}{j}\sum_{m=0}^j\tbinom{j}{m}^3\,\frac{140(561k+53)-13\color{blue}\alpha/4}{(2\cdot140^2+4\color{blue}\alpha)^k}$$
Level 10
$$\frac1{\pi}=\frac{16\sqrt{5}}{\sqrt{19}(76^2+4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}k\sum_{j=0}^k \color{blue}\alpha^{k-j}\tbinom{k}{j}\tbinom{2j}{j}^{-1}\sum_{m=0}^j\tbinom{j}{m}^4\,\frac{19^2(408k+47)-157\color{blue}\alpha/4}{(76^2+4\color{blue}\alpha)^k}$$
P.S. However, I do not have a rigorous proof for these families and the relevant literature do not seem to address general $\alpha$.
Note: The level $\color{red}m$ is given by an eta quotient $\frac{\eta(\tau)}{\eta(\color{red}m\,\tau)}$. For example, define
$$\lambda(\tau) =-6+\big(\tfrac{\eta^2(3\tau)}{\eta(\tau)\,\eta(\color{red}9\tau)}\big)^6 -27 \big(\tfrac{\eta(\tau)\,\eta(\color{red}9\tau)}{\eta^2(3\tau)}\big)^6$$
so $\lambda\Big(\tfrac{3+\sqrt{-163}}{6}\Big)=-640320$ is level $9$.
(Per S. Cooper's request.)
I. Table relating level $1$ with level $9$.
The general form apparently is,
$$\frac{1}{\pi}=\frac{12}{(C)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k} \tbinom{3k}{k}\tbinom{6k}{3k} \frac{\color{red}3A\,k+B}{(-C^3)^k}$$ and, $$\frac{1}{\pi}=\frac{12}{(C-4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{3j}\tbinom{2j}{j}\tbinom{3j}{j} \frac{A\,k+B+N\color{blue}\alpha/3}{(-C+4\color{blue}\alpha)^k}$$
for general real $\color{blue}\alpha$ and where,
$$\begin{array}{|c|c|c|c|c|} \hline d&A&B&C&N\\ \hline 11&154/9&5&32&4/3\\ 19&114&25&96&4\\ 43 &5418 &789 &960 &24\\ 67 &87234 &10177 &5280 &76\\ 163 &181713378 &13591409 &640320 &1448\\ \hline \end{array}$$
The variables $A,B,C$ are known to have closed-form expressions in terms of $d$. Presumably $N$ should have as well.
P.S. Note also the equivalent forms,
$$s_k(\color{blue}\alpha)=\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{3j}\tbinom{2j}{j}\tbinom{3j}{j}=\sum_{j=0}^k \color{blue}\alpha^{k-3j} \tbinom{k}{j}\tbinom{k-j}{j}\tbinom{k-2j}{j}$$
where the latter form is used in H. Chan and S. Cooper's paper "Rational analogues of Ramanujan's series for 1/π". The case $\alpha=-3$,
$$s_k(-3) = 1, -3, 9, -21, 9, 297, -2421$$
is one of the six sporadic sequences studied by Zagier and Cooper.
II. Table relating level $2$ with level $8$.
The general form apparently is,
$$\frac{1}{\pi}=\frac{192\sqrt{2}}{(C)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k} \tbinom{2k}{k}\tbinom{4k}{2k} \frac{\color{red}2A\,k+B}{(C^2)^k}$$ and, $$\frac{1}{\pi}=\frac{192\sqrt{2}}{(C+4\color{blue}\alpha)^{3/2}}\sum_{k=0}^\infty\tbinom{2k}{k}\sum_{j=0}^k \color{blue}\alpha^{k-2j} \tbinom{k}{2j}\tbinom{2j}{j}\tbinom{2j}{j} \frac{A\,k+B-M\color{blue}\alpha/4}{(C+4\color{blue}\alpha)^k}$$
for general real $\color{blue}\alpha$ and where,
$$\begin{array}{|c|c|c|c|c|} \hline d&A&B&C&M\\ \hline 6&\sqrt2&\sqrt2/4&(4\sqrt3)^2&\sqrt2/12\\ 10&10&2&12^2&1/3\\ 18 &70\sqrt6 &21\sqrt6/2 &28^2 &\sqrt6/2\\ 22 &385\sqrt2 &209\sqrt2/4 &(12\sqrt{11})^2 &17\sqrt2/12\\ 58 &870870&72798&396^2&37\\ \hline \end{array}$$