Easiest way to deal with this would be to have the mutiny partially succeed.... for Barnabus. Pick some of the pirate crew to be caught attempting to mutiny with the PCs, sink the Man's Promise, and give the caught mutineers a single lifeboat and maroon them in the open waters with no food, water, supplies, etc.
The boat eventually drifts near enough to Bonewrack Isle to boatwreck them there so to speak. There they can continue with the adventure, find another wrecked but salvageable ship similar to the Promise (maybe a Chelish navy vessel that would still need to be squibbed properly in book 2) and continue from there.
I don't like killing PCs unless they were being stupid. The goal was to mutiny, just not that early. I would have Barnabus set an example as you stated and likely kill their favorite NPC to stay within his character. You can replace this NPC on Bonewrack (maybe a previously marooned sailor from the new wreck they find), or wait till they get the boat into dock in book 2 for the squib treatment.
While it's understandable for the party to attempt to take advantage of the opportunity here (steal the faster ship during a battle, get away before Barnabas and company can react), clearly the timing for the mutiny isn't right and the party really has the deck stacked against them for being level 1 or 2 against a party of mid-to-high level antagonists.
If you're going to have Peppery get involved (as it's natural she would do something about this), I would definitely go with Ice Storm as it's the lowest damage alternative (5d6 total) but will cover much of the ship and is probably the least damaging to the ship itself (Barnabas isn't going to like fireballs blowing holes in the deck of his own ship). You also get to the play the "(ice) rocks fall, everyone dies" trope, with the benefit that hopefully everybody important survives it, albeit probably in the negatives. Just hand-wave the stabilization rolls and assume Barnabas manages to get someone over to the boat in time to preserve all of the PCs unless they're negative Con and outright dead. Then they all get woken up and you have Barnabas go Neegan on them and the surviving NPCs.
Playing off of the above suggestions, I'd have the party's favorite/most friendly NPC speak up and state that they were the mutiny leader, it was all their idea, thus taking the heat from the PCs. Then, showcasing Barnabas' extreme sadistic side, rather than just kill the NPC, even with something gruesome as a keelhauling, he beats them up and cripples them somehow (lose of an arm/leg always a good piratey choice, or a slash across the eyes to blind them. You want something more permanent here than straight HP damage that the party cleric can just fix in 1 minute or less, and to serve as a visible reminder of the price for crossing Barnabas and failing) before throwing them overboard (in shark infested waters?) while announcing to the PCs "well, are you going to go follow your leader then?" and soliciting/threatening/forcing them overboard as well along with the other NPCs you want along for the Bonewrack Isle adventure. Then they either all wash up on Bonewrack or Barnabas leaves them a small lifeboat so they can starve to death at sea rather than get eaten by the sharks (or so he thinks). This gives the PCs a chance to still save the favorite NPC, but it'll be an ongoing problem for them as keeping a crippled character safe on Bonewrack Isle is going to be quite the feat (particularly if they're also the unlucky bastard who gets captured by the grindyglows because the DM is just as sadistic as Barnabas. ;) ). If you still want the munity portion of the adventure/offer the players some revenge, you can have them discover the Man's Promise with Plugg and Scourge and friends aboard it, sheltering on the island after they also got blown off-course in the storm and needing to make repairs (instead of the abandoned ship).
Another plus to having the Man's Promise show up off-shore is that you can split up the friendly NPCs here. Some end up getting thrown overboard with the PCs, and they have them available during the Bonewrack Isle segment. The rest are part of Scourge and Plugg's crew on the Man's Promise and need to be rescued (because Barnabas can't kill ALL of the mutineers, he needs some manpower left), but Scourge and Plugg have definitely been giving those NPCs the sharp end of the stick and making their lives miserable, with the expectation that once the Man's Promise is in port and they aren't needed anymore, they're all dead too. Basically slowly torturing them to death, just with more steps, while getting some work out of them while tying/locking them up at night.
Another option with Peppery is to quietly adjust her spell selection so that she has Summon Monster ready. I believe there's an event in the race from Book Three where she summons a Water elemental to slow the party's ship down with its whirlpool ability. She could use the same trick here to keep the Wormwood from escaping, as a non-lethal method of helping Barnabas reclaim his ship. Of course, that then raises the issue of what happens if the PCs still refuse to surrender, but you've always got that Ice Storm in your back pocket I suppose as a follow-up if they're still stubbornly refusing to concede. And Barnabas probably won't care much about taking 5d6 damage if it means everyone around him "dies". This will also let them know of the water elemental trick ahead of time, so they can try to think ahead and prepare for it in Book Three.