The following is not an answer, since the last comment to your question already answers most of it.

It is, rather, a "better question".

What you really want to ask is: let $f: X\to\text{Spec}(k)$ be a proper, geometrically connected, geometrically reduced scheme over a perfect field.

For integers $p,q$, $p\ge 0$, does there exist a $k$-group scheme $P^{p,q}_{X/k}$?, such that:

 - $P^{p,q}_{X/k}$ is locally of finite type over $k$ and quasi-separated.
 - The component of the identity $(P^{p,q}_{X/k})^0$ is connected, finite type and separated.
 - Formation of $P^{p,q}_{X/k}$ commutes with separable field extensions on $k$.
 - If $X$ is smooth over $k$, then $(P^{p,q}_{X/k})^0$ is smooth.
 - If $X$ is smooth and projective over $k$, then $\text{NS}^{p,q}(X) := P^{p,q}_{X/k}/(P^{p,q}_{X/k})^0$ is a constant étale $k$-group scheme whose value group is finitely generated.
 - If $X$ is smooth and projective over $k$, and $k$ is a subfield of $\mathbf{C}$, then
 $$(P^{p,q}_{X/k}\times_k\mathbf{C})(\mathbf{C}) = H^p_{\mathcal{D}}(X_{\mathbf{C}}^{\rm an},\mathbf{Z}(q)),\ \ ((P^{p,q}_{X/k})^0\times_k\mathbf{C})(\mathbf{C}) = J^{p,q}(X^{\rm an}_{\mathbf{C}}/\mathbf{C}),\ \ (\text{NS}^{p,q}(X)\times_k\mathbf{C})(\mathbf{C}) = \text{Hdg}^{p,q}(X_{\mathbf{C}}^{\rm an}/\mathbf{C}).$$

In other words, you're actually looking for "higher Picard functors", representable under the assumptions stated at the beginning of this post, such that their connected component of the identity is an extension of a connected smooth affine $k$-group scheme by an abelian $k$-variety, and such that an analog of the theorem of the base holds true.

For $q=1$, this problem is related to the question whether $R^{p-1}f_{\rm fppf, *}\mathbf{G}_m$ is representable, which, for $p=2$, we now to be the case by work of Artin. Even the case $q=1$, for arbitrary $p\ge 2$, is extremely hard. A seemingly easier question is whether $R^{p-1}f_{\rm fppf, *}\mu_n$ is representable, and even this question (posed by Artin when $n$ is prime) is very hard too, and answered positively only in very special cases (mostly by work of Lieblich).

If the above dream were true, then your question becomes indeed very interesting, if asked about $(P^{p,q}_{X/k})^0(k^{\rm sep})^{\text{Gal}(k^{\rm sep}/k)}$ instead of $J^{p,q}(X_{\mathbf{C}}/\mathbf{C})$, and when $k$ is a finitely generated field.

The functors $P^{p,q}_{X/k}$, which I'm not going to define here, occurred at some point in my work, where I was able to show their representability in some special cases I was interested in (by direct construction, ie. not via Artin's axioms). The general question, open for now, seems to be very hard (if true at all).