The spring should only have a slight amount of tension to allow for tree growth without damaging the sign or the bark of the tree. It is important that as the tree grows and the spring compresses that you check on the mounting and back out the screw as needed to give some space in the spring compression.
If you did want to use the smaller tags with the tree mount hardware, consider buying tags without the pre-drilled holes on the left and drill your own hole in the top center. If you do this, just pick an appropriate template that will allow for placement of the hole where you want and where it will not be located in the image or text area. If you have someone with a drill press, it would be an easy job.