Given an integer partition $\lambda$, denote $\ell(\lambda)=$ length, $\vert\lambda\vert=$ size and $\lambda=$ conjugate of $\lambda$. Allow to write $\lambda\vdash n$ either as $(\lambda_1,\dots,\lambda_{\ell})$ or else as $1^{m_1}2^{m_2}\cdots n^{m_n}$.
$\DeclareMathOperator\Ev{Ev}$Define $\Ev(\lambda)={}$the set of all partitions obtained by replacing each $\lambda_i$ with either $2\lambda_i$ (doubling) or $\lambda_i,\lambda_i$ (two copies) and \begin{align*} \mathcal{R}_N(2\vert\lambda\vert)&:=\{\mu \vdash 2\vert\lambda\vert: \,\, \ell(\mu)\leq N; \, \text{$\mu_i$ is even for all $i$}\}, \\ \mathcal{R}^c_N(2\vert\lambda\vert)&:=\{\mu \vdash 2\vert\lambda\vert: \,\, \ell(\mu)\leq N; \, \text{$\mu_i'$ is even for all $i$}\}, \\ \chi_{\lambda}^{\mu}&:= \text{the character of the symmetric group $\mathfrak{S}_{\vert\lambda\vert}$ for $\vert\mu\vert=\vert\lambda\vert$}, \\ z_{\lambda}&:=\prod_{j\geq1}m_j!\,j^{m_j}. \end{align*} ${\color{blue}{\textit{Caveat}}}$: the multiset $\Ev(\lambda)$ has $2^{\ell(\lambda)}$ elements and multiple Young diagrams are allowed.
(${\color{red}{\textit{Updated}}}$) MILDER QUESTION. Is this true? For any $N\geq1$, we have that $$\sum_{\lambda\vdash n} \frac1{z_{\lambda}} \, \sum_{\tilde{\lambda}\in \Ev(\lambda)} (-1)^{\ell(\tilde{\lambda})} \sum_{\mu\, \in \,\mathcal{R}_{2N+1}(2\vert\lambda\vert)} \chi_{\tilde{\lambda}}^{\mu} =\sum_{\lambda\vdash n} \frac1{z_{\lambda}} \, \sum_{\tilde{\lambda}\in \Ev(\lambda)}\,\, \sum_{\mu\, \in\, \mathcal{R}_{2N}^c(2\vert\lambda\vert)} \chi_{\tilde{\lambda}}^{\mu}. $$
Example. If $\lambda=111$ then $\Ev(111)=\{222,2211,2211,2211,21111,21111,21111,111111\}$. If, in addition, $N=1$ then $\mathcal{R}_2^c(2\vert111\vert)=\{33\}$ while $\mathcal{R}_3(2\vert111\vert)=\{6, 42, 222\}$.
Remark. The limiting case $N\rightarrow\infty$ is true. Indeed, this follows since the elements of $\mathcal{R}_{\infty}(a)$ and $\mathcal{R}_{\infty}^c(a)$ are transposes of each other. In such event, it is clear that $$\chi_{\tilde{\lambda}}^{{\color{red}\mu'}} =(-1)^{\vert\tilde{\lambda}\vert+\ell(\tilde{\lambda})}\cdot \chi_{\tilde{\lambda}}^{{\color{red}\mu}}.$$
