Generalizing two infinite products for $\operatorname{sinc}(x)$ and their 'dual' infinite product

Using very truncated series $$a_m=\frac1m{\sin \left(x m^{1-k}\right) \csc \left(x m^{-k}\right)}$$ $$a_m=1-\frac{1}{6} \left(m^2-1\right) m^{-2 k}x^2+\frac{1}{360} \left(3 m^4-10 m^2+7\right) m^{-4 k}x^4+O\left(x^6\right)$$

Taking logarithms and expanding again $$\log(a_m)=-\frac{1}{6} \left(m^2-1\right) m^{-2 k}x^2-\frac{1}{180} \left(m^4-1\right) m^{-4 k}x^4+O\left(x^6\right)$$ $$\sum_{m=1}^\infty \log(a_m)=-\frac{\zeta (2 k-2)-\zeta (2k)}{6} x^2 -\frac{\zeta (4 k-4)-\zeta (4 k)}{180} x^4 +O\left(x^6\right)$$ $$\color{red}{\prod_{m=1}^\infty a_m \sim \exp\Bigg[-\frac{\zeta (2 k-2)-\zeta (2k)}{6} x^2 -\frac{\zeta (4 k-4)-\zeta (4 k)}{180} x^4+\cdots \Bigg]}$$ This produces quite well the shape of your curves which in fact are gaussian.