Nonttehless, Nappy still held out for the surrender of Venetia as the price for his neutrality in an Austro-Prussian War. See AJP Taylor's Struggle For Mastery In Europe for an account of the diplomacy.
I do not doubt your word on what Taylor wrote on his book, but if he wrote so, he just wrote something wrong.
The fact is a bit more complex.
As I said Nap3 had bargained the annection of Nice Savoy to France in exchange for the annection of Lombardy-Venice to Pidemont.
He grabbed Nice-Savoy in '59, but he did not fullfil his part of the contract, thus having them
de facto but not
de iure (in practice, they were militarly occupied territories).
Nap3 would probably be glad of some 1866-war outcome giving Venetia to Pidemont since that would make his position a little more legitimate on Savoy-Nice, but did not make any bargain about him.
It was Franz Josef, instead, that asked him to be a mediator.
Note that this was
not because Nap3 could be viewed as a Italy-friendly power, but quite the opposite, because Italy at the time was hostile to him.
In practice, the figure of him as a mediator was a calculated insult that Franz Josef threw in the face of Vittorio Emmanuele, to stress the fact that Austria was not defeated by italian arms, and in order to get Venice, Italy had to gulp down its pride and bargain with France (which Italy was in bad terms with)