My idea is that there is a hidden pirate witch on board that can get a lightning bird to attack once a day, or something, but that she needs to burn incense as a consumable part of the spell.
This is actually exactly just what you need already. Monsters and adversaries in 5E do not need to follow exactly the same rules as PCs, and in fact in many cases explicitly don't. There's no reason to not just have exactly what you describe to create the experience and challenge you want for your game.
From the introduction to the Monster Manual:
A monster is defined as any creature that can be interacted with and potentially fought and killed. Even something as harmless as a frog or as benevolent as a unicorn is a monster by this definition. The term also applies to humans, elves, dwarves, and other civilized folk who might be friends or rivals to the player characters.
So, in your scenario, "Pirate Witch" is a "monster" by definition. Then, the DMG has a whole section Creating a Monster, which starts out like this:
The Monster Manual contains hundreds of ready-to-play monsters, but it doesn’t include every monster that you can imagine. Part of the D&D experience is the simple joy of creating new monsters and customizing existing ones, if for no other reason than to surprise and delight your players with something they’ve never faced before.
The first step in the process is coming up with the concept for your monster. What makes it unique? Where does it live? What role do you want it to serve in your adventure, your campaign, or your world? What does it look like? Does it have any weird abilities? Once you have the answers to these questions, you can start figuring out how to represent your monster in the game.
So, this is pretty simple: you've already got a great idea for what this monster is (it hides, and it as an affinity for "lightning birds").
You may decide that this is an intrinsic ability of the monster itself (where "summon lightning bird" is action). The "Creating a Monster" section has great guidance on filling out that idea into something you can use in your game.
Or you could simply use an existing witch-like monster as a template, and make that and lightning birds simply working together (like you might have an tribe of orcs with trained wolves). In that case, look to the DMG's guidance on Encounter Design for advice on putting it all together.
I've never created a monster with this particular idea, but I've certainly done similar. In one case, I had a villainous necromancer who used mushroom-people as minions — and since I had this idea on the fly in the middle of a game, I used the Necromancer from the Monster Manual just as-is but described him as having rotten, moldy appearance (the party was suitably grossed out) and grabbed some myconids as the minions. Had I had more time to work this out, I might have come up with a custom necromancer with rot-and-decay focused spells. Either way can work.
The key thing is that the Monster Manual is meant to be inspiration and a quick, ready-to-use source, but isn't intended to be comprehensive. In fact, in my experience, especially since I've played enough D&D that there are few surprises left in the book, it's super-fun when the DM takes the effort to create a unique opponent that fits the story, like you are doing here.