I seem to remember the orientation axis not mattering in Unity if the rig was created in front viewport. Its worth a try to troubleshoot.
Also consider parenting the root to a dummy oriented correctly before export.
Also Parenting the root to an empty game object in Unity.
Have you googled “export cat rig to unity”?
Seems to have a lot of info about it.
Glad you found a solution that didn’t require re-rigging/re-skinning - having to do that is SO demoralizing.
Looking forward to seeing some of your animations.