Peter, first, Trenitalia does not accept U.S. credit cards. That's why you are getting this message. Unfortunately, they don't give you the correct reason. Second, for your journey to La Spezia, you will be on at least two different trains. Most often, these will Regionale trains. No seat reservations are permitted on this class of trains. It like you local bus system on tracks. Also, Trenitalia often does not sell Regionale that far out in advance.
Don't worry about this travel. It will not be sold out. The tickets on Regionale trains are like your local bus tickets. The tickets are good for 60 days and that's the only date you will see on the tickets. No train number or departure time. That's the Regionale ticket. As long as your departure point is Florence and your destination is La Spezia Centrale, you can ride any Regionale train making that run for 60 days. Two big secrets: 1)get there a bit early to make sure that you are among the first on and the first to find a seat. 2) you can't sell out a Regionale train so the runs you see are not sold out. If no seats, you can stand. Since Trenitalia doesn't put train numbers on their Regionale tickets and they don't know which train you will be on, they will sell you the ticket. Wait until you get to the CT and see the trains and crowds there.