Add shared folder to AlwaysOn AG Failover
The recommended way to do this is the way you don't want to do it--by creating a shared disk and the necessary cluster resources for a file share.

Other cluster options would require some sort of replication functionality.  Carbonite  Availability GeoCluster will allow you to create software replicated disk resource to accomplish this, and there are some other products that will do the same thing.  However, using third-party products introduces some risk because if there is a defect your cluster group may fail over or even fail to come online.  Accordingly, I personally would not opt for one of these solutions.