Analysis of the function $\prod_{n=-\infty}^{ \infty }(1-e^{-a{(x+n)^2} })$
It's not an answer, but there are a lot of graphs which can't be put into a comment.
The function $\prod\limits_{n=-\infty}^{ \infty }(1-e^{-a{(x+n)^2}})$ is very close to $C \sin^2(\pi x)$ for $a \leq 1$, as can be seen by numerical experiments:
1) For $a=1$ the scaling constant $C=0.0390070548953618...$ and the functions are practically indistinguishable.
2) For $a<<1$ the functions are also very close with appropriate scaling.
3) However, for $a>>1$ the functions are different, since $f(x)$ approaches a constant around its maxima.
The overall $a$ dependence can be seen in the following 3D graph and the $f(1/2)$ graphs for the dependence around a maximum. The latter has the character of a logistic function.
I didn't have much luck with analytics so far, but I will expand my post into a proper answer if something useful comes up.
Update. As asked by the OP, I tested their approximation, it's not very good:
