This is my first post here, Found the site through searching for German railroad lines during WWI on google.
Anyway...
Idea wouldn't work because Austria-Hungary is already in a state of deterioration by the 1880's. It doesn't have the leadership, will, or reasons to start an attack on Italy. Besides, why would it when it has enough trouble keeping peace within its own borders? (Balkans)
Another note: Austria Hungary was on somewhat good terms with Italy, evidenced by their formation of the Triple Alliance along with Germany in 1882.
Another complication: Germans and Austria Hungary are not Catholic. The Pope is. Why would they push for his interests?
Bismarck would also not support the war as it goes against his ideologies. The same ideologies that opposed the inclusion of Alsace and Lorraine after the Franco-Prussian War in 1870. He believed that in order to keep the nation strong, the people must be Germans. Alsace and Lorraine were French. It was Wilhelm I who wanted those territories. Now, If Austria Hungary takes over Italy, they'll have Italians in their territory, and according to Bismarck, that would theoretically make them a more unstable nation. Would Bismarck want his ally to become more unstable than it was?