Typically, when you are calculating the emission from a given source, the output will be a time-dependent current. You wrote this as $\mathcal{J}(t)$, but I will rewrite this as $\mathbf J(t)$ in order to emphasize that the current (or other such quantity, such as the derivative of a dipole moment) is a vector quantity.
As you rightly point out, if we want to calculate the harmonic spectrum, the first thing to do is to take a Fourier transform, i.e., we write
$$
\tilde{\mathbf{J}}(\omega)
=
\frac{1}{\sqrt{2\pi}} \int \mathbf J(t) e^{i\omega t} \mathrm dt
$$
in order to get the (vector) spectral amplitude $\tilde{\mathbf{J}}(\omega)$ of the current, and then we know that we can decompose the time-dependent current as a superposition of monochromatic oscillations in the form
$$
\mathbf J(t)
=
\frac{1}{\sqrt{2\pi}} \int \tilde{\mathbf{J}}(\omega) e^{-i\omega t} \mathrm d\omega
.
$$
So, what about the helicity? Well, that only makes sense on a harmonic-per-harmonic basis, because the full time-dependent $\mathbf J(t)$ might have a rather more complicated shape than just an ellipse. But if you filter down to the contribution of just one of the monochromatic oscillations,$$
\mathbf J_\omega(t)
=
\operatorname{Re}\mathopen{}\left[
\tilde{\mathbf{J}}(\omega) e^{-i\omega t}
\right]\mathclose{}
=
\frac12\left[
\tilde{\mathbf{J}}(\omega) e^{-i\omega t}
+ \tilde{\mathbf{J}}(-\omega) e^{+i\omega t}
\right]
$$
(where necessarily $\tilde{\mathbf{J}}(-\omega) = \tilde{\mathbf{J}}(\omega)^*$ to keep $\mathbf J(t)$ real-valued), then this is a vector-valued monochromatic wave which can indeed be elliptical.
So, this is where we define the ellipticity (and, with it, the helicity), by taking the components of this vector oscillation along the right- and left-handed circular basis. Thus, we define
\begin{align}
I_\circlearrowright(\omega) & = |\hat{\mathbf{e}}_\circlearrowright \cdot \tilde{\mathbf{J}}(\omega)|^2 \\
I_\circlearrowleft(\omega) & = |\hat{\mathbf{e}}_\circlearrowleft\cdot \tilde{\mathbf{J}}(\omega)|^2,
\end{align}
where $\hat{\mathbf{e}}_\circlearrowright = \tfrac{1}{\sqrt{2}}(\hat{\mathbf{e}}_x+ i\hat{\mathbf{e}}_y)$ and $\hat{\mathbf{e}}_\circlearrowleft = \tfrac{1}{\sqrt{2}}( \hat{\mathbf{e}}_x- i\hat{\mathbf{e}}_y)$. These are the quantities used to define the spectral ellipticity (or spectral helicity) as
$$
\varepsilon(\omega)
=
\frac{I_{\circlearrowright}(\omega) - I_{\circlearrowleft}(\omega)}{I_{\circlearrowright}(\omega) + I_{\circlearrowleft}(\omega)}
.
$$
(And, finally, about the name: there isn't a particularly strongly accepted convention about what to call ellipticity / signed ellipticity / helicity. Just use one that feels right to you, but use it consistently. My approach would be to call the above quantity the ellipticity, allowing it to be both positive and negative, and then denote the helicity as $h=\operatorname{sgn}(\varepsilon)$. But there's plenty of other possible approaches!)
