Converting a Bernoulli to a Gaussian
Suppose you had such a randomized procedure that takes a value in $\{-1,1\}$ and outputs a real number. Let $P$ and $Q$ be the output distribution on input $+1$ and $-1$ respectively.
Consider the extreme case of $\mu = +1$. In this case $Y = +1$ for sure, and you are outputting a sample from $P$, which means that $P$ should be an $\mathcal{N}(\nu, 1)$ distribution for some $\nu$ with $|\nu|$ being $\Omega(1)$. Similarly, the other extreme of $\mu=-1$ tells us that $Q$ is a $\mathcal{N}(\nu', 1)$ distribution for some $\nu'$ with $|\nu'|$ being $\Omega(1)$.
But now, for $\mu=0$, you are outputting the uniform mixture of two normals, which cannot be a normal distribution unless $\nu=\nu'$. In the latter case, you get $\mathcal{N}(\nu, 1)$ for some $\nu$ bounded away from zero.
Thus you cannot satisfy this requirement simultaneously for $\mu\in \{-1, 0, 1\}$.