OK, this will seem gross, but it does work! Have your hubby or handyman, whichever is willing, go pee on the fig tree and the surrounding ground. Mark the territory. Or, if he is a shy guy, have him pee in a bucket, soak rags in there and hang them on the tree.
This will discourage a ton of animals! How do I know? We have this apple tree with amazingly crisp, juicy wonderful apples waay down on the property, and the apples that were ripening all had bite marks (tests for ripeness from the critters,) so JD went down there unwillingly, and left his mark! A bounty of untouched apples was ours, and they are still (they were legion) in both fridges waiting for me to make pies and cobblers.
Just wash the fruit good! LOL!
Suzi