In light of the paper referenced in Tim Perutz' comment, I believe to have written down a desired map:
An oriented 2-plane field $\xi$ is equivalent to a pair $(\mathfrak{s},\phi)$ on $Y$, where $\mathfrak{s}$ is a spin-c structure and $\phi$ is unit-length spinor. By Proposition 28.1.2 (of Kronheimer-Mrowka's textbook), there exists an oriented 4-manifold $X$ with $\partial X=Y$ and carrying a spin-c structure $\mathfrak{s}_X=(S^+,\rho_X)$ which extends $\mathfrak{s}$, i.e. $\mathfrak{s}\cong(S^+|_Y,\rho_Y)$. Now the relative Euler class $e(S^+,\phi)$ satisfies $e(S^+,\phi)[X,\partial X]$ = $gr_z(X,\mathfrak{s}_X,[a])\in\mathbb{Z}$ for some configuration point $[a]$ associated to $\phi$, and this index is independent of $z$. And by Proposition 28.2.2, this is independent of the choice of $X$ (up to homotopy of $\phi$). This is where the isomorphism $\mathbb{J}(Y)\cong \Theta(Y)$, $\xi\leftrightarrow [a]$, comes from. We can thus write down a map $f:\Theta(Y)\to\mathbb{Z}_2$ via $\xi\mapsto e(S^+,\phi)[X,\partial X]\;\text{mod}2$.
However, when we apply this to $Y=S^3$ we get the flipped even/odd decomposition of $\widehat{HM}(S^3)$. Thus we can simply change our $f$ by $Aut(\mathbb{Z}_2)$, i.e. adding a $1$, to get agreement.
I would still be interested in other maps which don't necessarily apply to SW-Floer theory.
