Upper bound for $f(x) := e^{-\delta x^2 }\frac{\sinh\left(\sqrt{\delta^2 x^4 -x^2}\right)}{\sqrt{\delta^2 x^4 -x^2}}$

Let $\delta >0$ and consider $j\in \mathbb{N}$ such that $\delta 2^j >\frac{1}{2\sqrt{2}}.$ I am looking for an upper bound of

$$f(x) := e^{-\delta x^2 }\frac{\sinh\left(\sqrt{\delta^2 x^4 -x^2}\right)}{\sqrt{\delta^2 x^4 -x^2}}$$on the interval $\left[2^{j-1}, 2^{j+1}\right]$ of the form

$$f(x) \leq g(\delta) 2^{-2j}$$such that $\lim_{\delta \to 0}g(\delta) = 0.$

Thank you for any hint.