I would think Mulberries wouldn't be poisonous to dogs because of all the other berries they can eat, blueberries, strawberries, cranberries, etc. Maybe the dogs that have got sick from them at too many?
The only experience that I have with them (besides knowing they are a huge mess) was to make my very first pie using mulberries (30 years ago).....it was a disaster! DH ate the pie filling...or at least tried to
Maybe your neighbors would agree to trimming the tree back?