You could export with the vertical movement, then nuke it at runtime, after loading the skeleton data. To do that, get the animation with SkeletonData findAnimation and iterate through the Animation timelines to find the TranslateTimeline for the root bone. That will be the one where TranslateTimeline boneIndex is 0. Finally, remove that timeline from Animation timelines.

If you're still a Unity user, removing it at load time would look like this:

SkeletonDataAsset skeletonDataAsset;
SkeletonData skeletonData = skeletonDataAsset.GetSkeletonData(true);

// For all animations
foreach (var animation in skeletonData.Animations) {
   int timelineIndex = -1;
   var timelines = animation.Timelines;

   // Look for the TranslateTimeline for the root bone.
   for (int i = 0, n = timelines.Count; i < n; i++) {
      var timeline = timelines.Items[i];
      var translateTimeline = timeline as TranslateTimeline;
      if (translateTimeline != null && translateTimeline.BoneIndex == 0) {
         timelineIndex = i;
         break;
      }
   }
   

   // Remove the TranslateTimeline if it's found.
   if (timelineIndex != -1)
      animation.Timelines.RemoveAt(timelineIndex);
}

In Spine-Unity for Spine 3.7, the new SkeletonDataModifierAsset system allows you to write this as an asset, and then just add it to your SkeletonDataAsset inspectors for the skeletons that you want it to be applied to.

What you need is a transform constrain with x,y component separated which is not available in current version.
I also faced the same problem when animating character jump, I ended up using two bones to control the X,Y of the jump which make sense if you want realistic gravity on Y while using different easing function on X. After the animation is done, just disable the X movement by using transform constraint to lock the position of X bone.