I think I may have something here. This post will be long, and I have several proposals to outline, so bear with me.
There is one way I can think of to limit player-controlled Biplanes enough that they're not a way to circumvent the team-based (and role-based) nature of the game. Conveniently enough, this approach seems to me like it would ALSO help with the problem of Gunners being frequently cast as second-class citizens in the game. It requires several features to be included with Biplane behaviour:
1. Fuel. In place of the altitude gauge in the cockpit, or opposite/alongside it, there will be a fuel gauge. Fuel will replenish while the Biplane is on the deck of the Carrier and unmanned/engine is off. When the engine is on at 0 thrust, fuel should be used at a minimal rate, and when it's at full throttle, you should burn through fuel relatively quickly. Also, after taking damage, the fuel gauge should gradually drop, with the rate of fuel loss increasing as the plane takes more hits.
2. Ammo. Biplanes will be pre-loaded with ONE LOAD of standard ammo. A Biplane pilot can reload it with ONE LOAD of each specialist ammo they have on hand. Reloading a biplane will waste any unused ammo. Pre-loading will replace the default ammo with the specialist ammo loaded (allowing a double-load of one ammo type). Special ammo that increases (or reduces) turn speed will apply to the plane's pitch and roll capabilities.
3. Movement. Biplanes could be controlled by using the mouse (or right stick on a controller) to control pitch and roll. Turning speed should be adjusted based on the equipped weapon (faster turn speed for weapon = faster pitch and roll speeds) as well as the above mention of amo. The plane's current throttle setting should also affect turning, with a higher throttle setting slowing your turns. Speed should likewise be based on the weapon's firing arc (tighter arc = lower top speed and acceleration). The throttle controls could work like the helm with R and F, except that with 0 thrust, you stall and the plane's nose drops when the speed falls too far. Also, instead of a reverse thrust option, you'll have a single step below 0 thrust for "engines off". when switching either way between 0 thrust and engines off, there should be a short cooldown before the throttle can be moved again.
It's important to note that I'm suggesting for ALL of the above to be used to balance and limit Biplanes, NOT for one or two of those limitations to be implemented. The above conditions would ensure that biplanes need to regularly return to the ship to repair, refuel and re-arm. It would also make Gunners valuable for their ability to carry twice as much ammo. As for dealing damage to Biplanes, I'm leaning towards saying they should have a TINY amount of hull HP with armour HP roughly equal to a heavy weapon. As with anything else, armour can be repaired (but only while the plane is safely landed on deck) but the small amount of hull would be permanent damage.
As for how the Carrier itself should work, I have a few possible ideas:
1. A normal 4-player airship, with only 2 engines and only 1 gun mounted on the ship itself (relies on Biplanes for damage output). Captain equips 3 light weapon slots - one for each Biplane, and the single turret-mounted weapon. The ship would be relatively durable, but slow, much like a Galleon. With only 2 engines and a relatively flat layout, it should be relatively easy to Engineer, but this is because you'd potentially be seeing 2 crew members off the ship at times, and being 1 player short would be the expected normal situation. Also, having only 2 Engines makes it more vulnerable to being disabled, and also means that a single engine taking damage will cause more trouble for the pilot (much like Squids with 4 engines being tricky to handle while under fire).
2.a) A special game mode for 3 vs. 3 battles with each fleet having 1 Carrier and 2 regular ships. These ships would be larger, and could hold 6 players, with 2 - 3 turrets, 3 or 4 engines and the 2 Biplanes. With the 4 vs. 4 display of 2 ships top and 2 bottom, you could merge the top pair into a single entry with 3 crew per column. The top-right player is the Captain, and sets up the ship's loadout, while the top-left player would be the Squadron Leader, and is responsible for setting the weapons up on the Biplanes. Using this method, the ship could be made more complex, with an Engineer (or anyone with a rebuild-capable tool) being required to attach fuel lines to the Biplane after landing (automatic ejection when the engine is started). In this mode, it would be like a team deathmatch, but played to 4 or 6 kills, and the Carrier counts as 2.
2.b) An alternative game mode could have one team with a Carrier and two escorts, and the other with 3 or 4 ships trying to take the Carrier down. Similar rules to the above, but maybe with more crew and possibly even 3 or 4 Biplanes. In this mode, Carriers could count for 2 or 3 kills, depending on the size of the ship, and 5 kills to win might be more practical (3 point Carrier + 2 regular ships).
3. In any game mode (possibly exclusing 2 vs. 2), any 2 ships could be combined as a Carrier if a pair of Captains agreed. The structure of the ship would be much like what's described in 2.a) above, with 6 crew. In this case, the pair of ships would lose a crew slot each, and this action couldn't be taken if there are more than 3 players on either ship. The space normally taken by the 4th crew member would be replaced with an indicator to mark which "other ship" is the other half of the Carrier, and possibly to identify whether the top slot is the Captain or the Squadron Leader. Top-left should take precedence as Captain with top-right as second choice, while bottom-right should take precedence as Squadron Leader with bottom-left as second choice. This creates an order of Captaining vs. Squadron Lead being arranged under the same basic principles as reading in most European languages.
NOTE: I may come back and edit this with alternatives/additional thoughts/spelling/grammar corrections. I'm heading out now and don't have time for my usual proof-reading. Hope it's mostly coherent.