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If Age of Sail era Ships were to come into regular contact with Polynesian outrigger and Catamaran ships like the Hōkūleʻa, and the people using the ships did not start using the foreign design of ships, how could they adapt their ships to compete in an age of sail setting including naval combat with things like cannons?
The design of the hulls and would likely have to be changed drastically to suit so how could they make the required changes while still staying true to the original design?
The other answer is saying that you can take advantage of upwind speed and manoeuvrability. It is true, but it will not compensate for the enormous firepower of larger monohull warships. Let’s take an example: The galleys were able to move fast even when there was no wind, they were able to go straight upwind if needed. They were much more manoeuvrable. However, in the battle of Cape Celidonia, 5 Spanish galleons (big mono-hull sailing ships) defeated 55 ottomans galleys. This is typical from this period, when large sail-only ships started to replace galleys on the seas.
Basically, during late sail-era, the more fire power you could carry, the more likely you were of winning.
You are not talking about galleys, but light multi-hulls. I believe however that the problem will be similar: your vessels are small. It is difficult to build a large catamaran or outrigger, because of their shape. A monohull warship is basically a big cylinder. Catamaran are two smaller cylinder with pieces attached between them. So the limitation that you have are:
The advantage were already covered in another answer:
In an open war between well-defined opponents, larger boats, even if they can't catch your catamarans, can come in range of cities and gun them from the sea. They could occupy harbours. They can go further in the open sea. They can carry more men and resources, in a safer way. My guess is that the catamarans are going to lose, not because they are sunk, but because the large ships are going to cut their harbours, resources and so on.
You ask if anything can be done in terms of hull design. I believe that the only way is making it bigger. But probably, while trying to make them bigger, they are going to lose some of the advantages (think weight and structure fragilization). Maybe the architects are going to reinvent mono-hulls?
So, as these boats won’t stand in front a large mono-hull vessel. Their strategy, in order to exploit their advantage at the maximum, is to play it guerrilla style.
This strategy is close to what were (and are) doing a lot of pirates.
A lateen rigged mulithull vessel such as Hōkūleʻa have straight advantage over the square rigged monohull designs the European powers used.
Choosing when to have your combat is one of the most important factors in "random encounters" in the period. Many of the Hornblower type stories include statements about larger and faster vessels and the risks of encountering one.
They also spend a lot of their time trying to gain the controlling position upwind of the vessel they're chasing. The lateen rig of the Polynesians on a catamaran hull with appropriate leeboard/centerboard can sail 10 to 15 degrees closer to the wind than a square rigged monohull.
In this case, even when smaller, it's still faster, and sailing closer to the wind, you get to choose your opponents, and when and where to fight or not.
Under normal (modern) circumstances I would say that a multihull vessel is slower to turn, but in this case you're again comparing a schooner-lateen rig with a square rigger. If you maintain this difference in rigging then you maintain faster maneuvering as square rigs lose all speed to tack and can end up going backwards.
It's entirely valid to scale catamarans up to quite large sizes without having to change the rigs all that much. If you have the people and you have the timber you can build them up to galleon scales. However one of the key differences you'll notice when scaling up a multihull when compared to a monohull, apart from the significantly higher complexity of construction, is that monohulls have larger holds relative to deck size. Multihulls have much larger decks but can't haul as much cargo. This leaves you in a situation of having greater naval effectiveness close to your home ports but limited ability to project power at greater ranges.
In classic "Ships of the Line" combat, the two forces formed two parallel lines, and then battled it out. This was driven by the reload time of the cannons - it took a fair amount of time to reload, so you wanted many cannons to maximize your firepower.
You generally did not want to approach your enemy head on - they showed you their side, which had many guns, and you showed them front of your ship which had few.
But (!) an outrigger is highly maneuverable and has a VERY small cross section when seen from head-on. An outrigger would be very hard to hit when it was a approaching head on.
So a potential modification could be: take a note from modern catamarans and place a platform between your main hull and your secondary hull. It only needs to be strong enough / large enough to hold your guns. This platform can be utterly destroyed by enemy fire, and all that happens is you loss the use of the guns on it.
Now, load up as many guns as you can fit on it, mainly arranged to face forward. Charge at the enemy.
Your hull has a very small cross section, and the enemy has a large cross section - you should score many more hits, on average. Once you've fired all your forward facing guns, use your maneuverability to turn about, and steer away. The non-forward facing guns provide some cover fire for your retreat.
You reload at your leisure, outside of firing range, while another outrigger rushes in to take your place. This creates a steady stream of accurate fire against your enemy, who often misses due to your small target size.
You probably want to keep this fight at the end of cannon range - it maximizes your advantage in target size and avoids downward shots onto the broad "catamaran deck" you've created. A good volley from a Ship of the Line against a smaller ship from close range would be devastating.
Gunboats have been used well into the Napoleonic era. An outrigger canoe could, in principle, function the same way.
