This is most likely a mechanical problem rather than a battery issue. On an AE-1, a mirror that stays up after the shutter fires can indicate the mirror is binding or getting locked in the mechanism. One possible cause mentioned is a worn or deteriorated mirror bumper.

If the film advance also becomes difficult or stops, there may be a broken piece inside interfering with the winding mechanism.

Because the camera still advances the film and resets the mirror when you wind, the camera likely needs service rather than a simple user adjustment. The safest recommendation is to have it professionally inspected and serviced, especially on an older film body where internal foam or plastic parts may have aged or failed.