Like the French, the Italians had 70,000 tons of allocated battleship tonnage that they could use for building. Both countries drew up designs for 17,500 ton, 23,000 ton and 35,000 ton ships as those figures divided into the 70k gave 4, 3, and 2 ships respectively. Both countries found that designing ships at 17,500 tons gave not much more than heavily armed cruisers.
However once the Germans leaked news of the Deutschland class ships, Italy revisited the idea of a 17,500 ton 'pocket battleship'. As part of the rejuvenation of the fleet, the old battleship Leonardo da Vinci had been raised and the turrets removed and refurbished as spares for the four extant battleships. Plans were in the process of being formulated for the rebuilding of the old battleships, part of which was the removal of the centre triple turret from each ship. This would give the Italians 7 triple 12" turrets that could be used to construct 3 'pocket battleship' sized ships.
So this takes us out to what the Italians were planning. The Italian African colonies had never produced the income that would assist Italy to achieve what it saw as its rightful place in the world. If Italy could dominate North Africa then its revenue would increase astronomically. To make this happen the Italians would have to remove the British and French influence from those regions. The Italians knew they would be unable to do this alone and thus forged the Pact of Steel with Germany. The hope was that the Germans would occupy the French and German forces, allowing the numerically superior Italian forces to sweep east and west from Libya, rolling up the disorganised French and British forces.
The navies part in all this was to intercept and harry any French and British reinforcements reaching the embattled regions. The Livorno class ships with their long range could operate out into the Indian and Atlantic Oceans, doing two jobs, intercepting reinforcements, and tying down forces and denying them to the main battlefront in the Mediterranean.
(Reading back through this, it does sound a bit implausible, but then again it is only an excuse to put up another drawing.)