Roman gentes were not monolithic entities. One cannot speak at all of how the gens Fulvia as a whole would've reacted to something, because the gens Fulvia was nothing more than a bunch of people sharing common ancestry. Each individual of the clan had his own opinions and there was no hierarchy over the gens as a whole. Even at the lowest levels of family organization we don't see strict, mafia-esque hierarchies--Quintus Cicero was a devout Caesarian from an early date, though his brother Marcus opposed Caesar as early as 63. As a larger example Caesar's cousin Lucius was a Caesarian partisan, but Lucius' son (also Lucius) was a rather vocal Pompeian, who fought under Cato at Thapsus and died soon after. Both Lucullus and Crassus were of the gens Licinia but they were bitter enemies. Gentes were not hiveminds under singular leadership, they were collections of independent individuals who just so happened to be related to each other. One cannot speak of what a member of the gens Fulvia would've thought because the gens Fulvia was not a political organization

Sardinia was taken during the Mercenary War in 238, along with Corsica. Mercenaries stationed in Sardinia revolted from Carthaginian domination, and the appeal of these mercenaries to Rome for aid prompted the Romans to undertake an expedition to Sardinia, though at first they refused to assist, respecting the treaty with the Carthaginians. There was never any question of actually becoming involved in the Mercenary War, the question was whether to respect the terms of the treaty or take advantage of them--in the end, the Romans technically did neither, as their brief declaration of war on Carthage in 238 technically got around both problems.

I can find no reference to any member of the gens Fulvia taking an active role in the events of 240-238. This does not mean there were no Fulvii involved--obviously, as one of the older nobilis plebeian families there were members of the gens involved, but none of them held magistracies at the time. A Fulvius Flaccus was consul in 237 and fought in Cisalpine Gaul, as well as undertaking another campaign (this time across the Po) during his second consulship in 224. The New Pauly says that this guy was magister equitum in 213, and that he opposed Scipio's motion to take the war into Africa in 205, but that has very little relevance to Sardinia in 238, and we have no idea of his thoughts on the matter. The Fulvii Flacci, however, were a recent addition to the nobiles, with their first senatorial ancestor only having held the consulship in 264. The older branches of the family were the Fulvii Centumali, Curvi and Paetini, but of these the closest member to hold magistracy in 239 was a Fulvius Centumalus, consul in 229--we know nothing of his opinion on the Mercenary War, though during his consulship he landed at Corcyra and celebrated a triumph for it. The consul in 238 was Sempronius Gracchus, and it was under him that the Sardinians were subjugated. The consuls of 239 were Valerius Falto and Mamilius Turrinus, and I can't find a relationship between either of them or the consuls of 238 with any member of the gens Fulvia, and I can find no evidence for the opinions of any member of the gens Fulvia on the Mercenary War