Good evening,
there are several points in your message.
1-Route and travel time.
If you want to drive from Nice to Livorno by road,
it will be very, very, very slow. In late May, I think you should expect an average speed of 40 km/h. Since, roughly, there are about 400 km,
you can see what that means: nearly 10 hours of driving. The only faster section is the last one: from the exit of Viareggio to Livorno.
The reason for all this: urbanization and terrain.
Unfortunately, on this route, the road doesn’t offer a decent alternative to the highway.
So I recommend taking the highway and doing the whole trip in one day to avoid traffic problems. A route planner will calculate the duration and toll costs for you. That would give you 3 nights near Livorno and 2 full days to explore the area.
2-If you opt for mobile-home accommodation, there are campsites north of Livorno (beaches toward Pisa) and also in Torre del Lago Puccini (20 km north of Pisa, but it’s farther from Livorno). You’ll need to check if they have rentals and if they offer stays shorter than a week at the end of May.
3-Sightseeing. I haven’t visited Livorno, so I don’t know if the city is worth it.
You’ll be about 20 km from Pisa, one of the most beautiful cities in Tuscany (and Italy).
In Pisa, there’s more than just the Leaning Tower and the Square of Miracles. Two days allow you to visit at a relaxed pace: in terms of monuments and museums, you’ll only see exceptional things (check out the "visittuscany" site ).
I’ll add two suggestions near Pisa if you want to get out of the city.
In Pontedera, on the site of the former factories, you’ll find the Piaggio Museum.
"La Vespa"

and "l’Ape"

are inseparable from mobility in 20th-century Italy. And scooters are back in style.
A completely different vibe from Pisa.
Between Pontedera and Pisa, off the village of Calci, you’ll find the Charterhouse of Pisa (Certosa di Pisa a Calci). This former monastery is a beautiful monument in a countryside setting, with two museums on different themes (you can choose).


Hope this gives you some ideas.