It is a theorem of Gromov and Lawson, also Schoen and Yau, that no closed orientable three-manifold which contains an aspherical factor in its prime decomposition can admit a metric of positive scalar curvature, see Theorem IV.6.18 of *Spin Geometry* by Lawson and Michelsohn. In particular, as the three-manifold you're interested in is aspherical, it does not admit a metric of positive scalar curvature (irrespective of the nature of the fibers).

In fact, thanks to the solution of the elliptisation conjecture, we now know that a closed orientable three-manifold admits a metric of positive scalar curvature if and only if its prime decomposition contains no aspherical factors.