Let $Z,Z'\in\{-1,1\}^n$ be two independent vectors of i.i.d. Rademacher r.v.'s, where $1\leq n \leq d$ are two integers ($d\gg 1$). I am trying to get an upper bound on $$ \mathbb{E}_{ZZ'}\left[ \exp\left({k\left(e^{\sum_{j=1}^n \mathrm{Erf}(\varepsilon Z_j /\sqrt{d})} -1\right)\left(e^{\sum_{j'=1}^n \mathrm{Erf}(\varepsilon Z'_{j'} /\sqrt{d})} -1\right)}\right) \right] \tag{$\dagger$} $$ where $t\geq 1$, $\varepsilon \in(0,1]$, $k\geq 1$ is an integer, and $\mathrm{Erf}$ is the error function. In particular, I am interested in the dependence on $n$: since $$ \mathrm{Erf}(x) \operatorname*{\sim}_{x\to\ 0} \frac{2}{\sqrt{\pi}}x $$ I expect that, for $n = n(d) \ll \sqrt{d}$, $(\dagger)$ should be roughly $$ \mathbb{E}_{ZZ'}\left[ \exp\left({\frac{4}{\pi}\frac{\varepsilon^2 k}{d}\sum_{j,j'} Z_jZ'_{j'}}\right) \right] \leq e^{C \frac{\varepsilon^4 k^2}{d^2}\cdot n^2} $$ (the last bound for $d$ big enough). But I'm unclear if the above qualitative behavior still holds for $n \gg \sqrt{d}$: do we still get that $n^2$ dependence in the exponent?

Note: I am actually interested in the quantity $$ \mathbb{E}_{ZZ'}\left[ \exp\left({k \left(\prod_{j=1}^n (1+\mathrm{Erf}(\varepsilon Z_j /\sqrt{d}))-1\right)\left(\prod_{j'=1}^n (1+\mathrm{Erf}(\varepsilon Z'_{j'} /\sqrt{d}))-1\right)}\right) \right] \tag{$\ddagger$} $$ (same questions), but $(\dagger)$ is similar and hopefully a bit cleaner to analyze.