When you say "fibering" do you means as a smooth fiber bundle? Smooth fiber bundles with fiber $F$ and base $B$ are classified by homotopy classes of maps from $B$ to the classifying space $B\mathrm{Diff}(F)$. If you case $F$ is a $3$-manifold so $\mathrm{Diff}(F)$ is somewhat better understood, see Hatcher's survey. The following two examples show what challenges exist.

If $F=S^3$, then by Hatcher's proof of the Smale conjecture, or by recent results of Bamler and Kleiner discussed here the group $\mathrm{Diff}(S^3)$ deformation retracts to $O(4)$. Then we just need to give a diffeomorphism classification of the total spaces of linear $S^3$-bundles over the surface $B$. In the orientable case (i.e., when the structure group is $SO(4)$ rather than $O(4)$) such bundles are completely classified in Classification of Oriented Sphere Bundles Over a $4$-Complex by Dold and Whitney, and the bundle is completely determined by its second Stiefel-Whitney class in $H^2(B:\mathbb Z_2)\cong\mathbb Z_2$. One still has to see whether the total spaces are non-diffeomorphic, which I suspect is true and probably can be done similarly to the paper of Melvin: 2-sphere bundles over compact surfaces.

If $F$ is closed orientable hyperbolic $3$-manifold, then by Gabai's proof of Smale conjecture $\mathrm{Diff}(F)$ deformation retracts to a finite group $G:=\mathrm{Iso}(F)$. This can also be done via Ricci flow (see the above mentioned preprint of Bamler-Kleiner). Thus the bundle is classified by the homotopy classes of maps from $B$ to $BG=K(G,1)$, i.e., by the set of conjugacy classes of homomorphisms from $\pi_1(B)$ to $G$. I am not sure if anyone classified their total spaces up to diffeomorphism, but e.g. if $G=1$, then there is only the trivial bundle.

When you say "fibering" do you means as a smooth fiber bundle? Smooth fiber bundles with fiber $F$ and base $B$ are classified by homotopy classes of maps from $B$ to the classifying space $B\mathrm{Diff}(F)$. If you case $F$ is a $3$-manifold so $\mathrm{Diff}(F)$ is somewhat better understood, see Hatcher's survey. The following two examples show what challenges exist.

If $F=S^3$, then by Hatcher's proof of the Smale conjecture, or by recent results of Bamler and Kleiner discussed here the group $\mathrm{Diff}(S^3)$ deformation retracts to $O(4)$. Then we just need to give a diffeomorphism classification of the total spaces of linear $S^3$-bundles over the surface $B$. In the orientable case (i.e., when the structure group is $SO(4)$ rather than $O(4)$) such bundles are completely classified in Classification of Oriented Sphere Bundles Over a $4$-Complex by Dold and Whitney, and the bundle is completely determined by its second Stiefel-Whitney class in $H^2(B:\mathbb Z_2)\cong\mathbb Z_2$. One still has to see whether the total spaces are non-diffeomorphic, which is true (Hint: from the long exact sequence of the (pair $5$-disk bundle over $B$, its boundary) note that the inclusion of the boundary is injective on the 2nd cohomology, so the 2nd Stiefel-Whitney class of the disk bundle has nonzero restriction of the boundary iff $w_2\neq 0$. Finally, $w_2$ is a homotopy invariant for closed manifolds, so the two sphere bundles are not even homotopy equivalent).

If $F$ is closed orientable hyperbolic $3$-manifold, then by Gabai's proof of Smale conjecture $\mathrm{Diff}(F)$ deformation retracts to a finite group $G:=\mathrm{Iso}(F)$. This can also be done via Ricci flow (see the above mentioned preprint of Bamler-Kleiner). Thus the bundle is classified by the homotopy classes of maps from $B$ to $BG=K(G,1)$, i.e., by the set of conjugacy classes of homomorphisms from $\pi_1(B)$ to $G$. I am not sure if anyone classified their total spaces up to diffeomorphism, but e.g. if $G=1$, then there is only the trivial bundle.

When you say "fibering" do you means as a smooth fiber bundle? Smooth fiber bundles with fiber $F$ and base $B$ are classified by homotopy classes of maps from $B$ to the classifying space $B\mathrm{Diff}(F)$. If you case $F$ is a $3$-manifold so $\mathrm{Diff}(F)$ is somewhat better understood, see Hatcher's survey. The following two examples show what challenges exist.

If $F=S^3$, then by Hatcher's proof of the Smale conjecture, or by recent results of Bamler and Kleiner discussed here the group $\mathrm{Diff}(S^3)$ deformation retracts to $O(3)$. Then we just need to give a diffeomorphism classification of the total spaces of linear $S^3$-bundles over the surface $B$. In the orientable case (i.e., when the structure group is $SO(3)$ rather than $O(3)$) such bundles are completely classified in Classification of Oriented Sphere Bundles Over a $4$-Complex by Dold and Whitney, and the bundle is completely determined by its second Stiefel-Whitney class in $H^2(B:\mathbb Z_2)\cong\mathbb Z_2$. One still has to see whether the total spaces are non-diffeomorphic, which I suspect is true and probably can be done similarly to the paper of Melvin: 2-sphere bundles over compact surfaces.

If $F$ is closed orientable hyperbolic $3$-manifold, then by Gabai's proof of Smale conjecture $\mathrm{Diff}(F)$ deformation retracts to a finite group $G:=\mathrm{Iso}(F)$. This can also be done via Ricci flow (see the above mentioned preprint of Bamler-Kleiner). Thus the bundle is classified by the homotopy classes of maps from $B$ to $BG=K(G,1)$, i.e., by the set of conjugacy classes of homomorphisms from $\pi_1(B)$ to $G$. I am not sure if anyone classified their total spaces up to diffeomorphism, but e.g. if $G=1$, then there is only the trivial bundle.

When you say "fibering" do you means as a smooth fiber bundle? Smooth fiber bundles with fiber $F$ and base $B$ are classified by homotopy classes of maps from $B$ to the classifying space $B\mathrm{Diff}(F)$. If you case $F$ is a $3$-manifold so $\mathrm{Diff}(F)$ is somewhat better understood, see Hatcher's survey. The following two examples show what challenges exist.

If $F=S^3$, then by Hatcher's proof of the Smale conjecture, or by recent results of Bamler and Kleiner discussed here the group $\mathrm{Diff}(S^3)$ deformation retracts to $O(4)$. Then we just need to give a diffeomorphism classification of the total spaces of linear $S^3$-bundles over the surface $B$. In the orientable case (i.e., when the structure group is $SO(4)$ rather than $O(4)$) such bundles are completely classified in Classification of Oriented Sphere Bundles Over a $4$-Complex by Dold and Whitney, and the bundle is completely determined by its second Stiefel-Whitney class in $H^2(B:\mathbb Z_2)\cong\mathbb Z_2$. One still has to see whether the total spaces are non-diffeomorphic, which I suspect is true and probably can be done similarly to the paper of Melvin: 2-sphere bundles over compact surfaces.

If $F$ is closed orientable hyperbolic $3$-manifold, then by Gabai's proof of Smale conjecture $\mathrm{Diff}(F)$ deformation retracts to a finite group $G:=\mathrm{Iso}(F)$. This can also be done via Ricci flow (see the above mentioned preprint of Bamler-Kleiner). Thus the bundle is classified by the homotopy classes of maps from $B$ to $BG=K(G,1)$, i.e., by the set of conjugacy classes of homomorphisms from $\pi_1(B)$ to $G$. I am not sure if anyone classified their total spaces up to diffeomorphism, but e.g. if $G=1$, then there is only the trivial bundle.