You can use the following tags together at the centre of the wildlife bridge (draw it as a line):
highway=path
surface=grass
bridge=yes
width=[Number of metres]
![highway=path with bridge][1]
And of course
barrier=fence
![barrier=fence][2]
for the wildlife protection.
And maybe you can use this tags for the bridge area:
natural=grassland
layer=1
![natural=grassland][3]
[1]: https://wiki.openstreetmap.org/w/images/7/7d/Rendering-highway_path_bridge_yes-mapnik.png
//wiki.openstreetmap.org/w/images/7/7d/Rendering-highway_path_bridge_yes-mapnik.png
[2]: http://help.openstreetmap.org/upfiles/204964.png
/upfiles/204964.png
[3]: http://help.openstreetmap.org/upfiles/120px-Natural-Grassland.png/upfiles/120px-Natural-Grassland.png