Because extended naval deployments are expensive, full blown wars are even more expensive and Britain is financially constrained after the Seven Years War (see Revolution, American). So the chain of events could be France invades, Britain blockades the island, full blown war is averted with compromise agreement. Neither side wants a war in 1768, but neither side wants to be humiliated either.
A blockade is
exactly the sort of thing that leads to a "full blown war," especially a blockade just off France's own coastline. After all, a blockade doesn't mean anything unless the blockaders are willing to fire upon the blockade-runners. One could argue that the Seven Years War (as a
global rather than a strictly American war) was itself sparked by British interdiction of French transports moving troops overseas; why should it be different this time? This would be an open provocation to war, an aggressive and hostile imposition of British demands into a bilateral Franco-Genoese matter. The time for Britain and France to stop this with a minimal lack of face is long past; the Fourth French Intervention, which eventually led to the conquest, started back in
1764. By 1768 there are already considerable French forces on the island who have been there for years. If Britain wanted to keep France out of Corsica they ought to have secured that as part of the peace that ended the Seven Years War. That might be possible, but it doesn't fulfill this WI (unless Britain for some reason subsequently decides to sell the island to Naples, which is... less than intuitive).
Make no mistake, if the French are stopped from invading Corsica by a British blockade and
back down, the humiliation will be enormous. Pawning off the island to Naples thereafter will not decrease that humiliation one iota. The world will see the British imposing themselves into French affairs and telling Louis what to do, and the French backing down in fear. Wars may be expensive, but France doesn't sell its honor cheaply.
The real issue here is that giving/selling the island to Naples doesn't
solve anything. It doesn't help the French, because their whole goal was to keep Corsica from falling into the hands of another power (including Spain and her associated states). It doesn't help Paoli, who will fight either way. It helps the British only
slightly, as while they get to poke France in the eye (which the British do love) the island is still going to a Bourbon power. It doesn't really even help the Neapolitans - Corsica produces almost nothing and the island is by this point famous for rebellion and intractability. It was a
huge money pit for France that never came close to paying for its own administration, let alone paying back the costs of invasion and occupation. That's something that seriously needs to be considered - does Ferdinand of Naples even
want this rock? France only annexed it to deny it to others, a concern which Ferdinand does not share.
If I had to pick the most promising way to do this I think I'd avert the French conquest entirely. Kill Paoli, get Corsica to fall back into civil war, or by some other means allow the status quo (that is, nominal but mostly ineffective Genoese control over the island) to continue longer. By 1769 the Revolution had been ongoing for 40 years, you can probably stretch it out a little more. Then, at some future point - perhaps during the American War of Independence, when the major powers are busy elsewhere - have the King of Naples step in and offer to take the island off Genoa's hands, while at the same time making deals with some of the Corsican factions/leaders so they'll accept the cession. Perhaps this is all masterminded by John Acton, the pro-British minister in Naples who promoted a naval buildup and maritime trade. The exhausted Genoese let Corsica go, the Corsicans (or at least some of them) say "well, autonomy within Naples is better than slavery to Genoa" and accept the cession, the British are reasonably pleased because Acton is their guy, and the French are occupied elsewhere and can't do much except protest. I still doubt Ferdinand actually
wants Corsica but perhaps Acton or other ministers convince him, and the island is perhaps a
little more attractive if the Corsicans (or at least some of them) actually support the cession, rather than France/Britain just handing Ferdinand a seething ball of rebellion and saying "here, have fun with this."