Just found this description on Bicitalia: perhaps my initial thoughts of following the Ciclopista del Sole were a tad premature as number 3 follows more closely the Via Francigena…
N°3 La Via dei Pellegrini (The Pilgrim Route)
This route is along the old Via Francigena as far as Rome (thus forming part of EuroVelo Route 5). It begins on the Swiss border at Chiasso/Como, then heads for Rome passing through Milan, Parma, Lucca, Siena. From Rome we propose to reach Brindisi (using long stretches of the Via Appia), which for centuries was the port of departure for pilgrims, crusaders and knights leaving for Jerusalem.
…and it does take you all the way to Brindisi. What was the logic following the Ciclopista del Sole in the first place apart from the attraction of cycling by the sea?