Oh, that's easy!

The size and speed of the bullet only matters when you're relying purely upon kinetic energy to take down the target.

However, if you have relatively tiny fairies who literally can't handle the recoil of a weapon with a decent kinetic energy, change the paradigm.

Instead of trying to make a gun capable of punching a lethal hole in what is essentially a kaiju to these fairies, make a gun capable of delivering a lethal toxin to their enemies.

There are many neuromuscular-blocking drugs such as curare, or other toxic agents such as strychnine that are relatively rapidly lethal that could be placed in lethal quantities within a projectile that would be able to penetrate human skin, yet would otherwise not carry enough energy to be lethal except with a very well-placed shot.

So, as long as a fairy can get a poisoned shot from a gun into a human's flesh, biochemistry will do the rest fairly quickly. For a small being who is far better placed to make the best of available cover than a human, taking a shot and retreating until the toxin has done its job makes far more sense than standing their ground and trying to take down a human with what amounts to the fairy-sized equivalent of elephant guns.

So, with toxins, a fairy can be lethal with what amounts to a fairy-sized revolver or a light rifle. They wouldn't need to use any particularly heavy weapons against regular humans.

That presupposes that humans don't go after fairies while wearing body armour. However, when an armoured-up human comes along, the fairies can break out their equivalent of crew-served artillery and deploy airborne toxins... but that's an answer to another question.

Additionally, once it becomes known that fairies use poisons, they'll become feared just for that. A fairy wouldn't even need to carry a gun to be able to poison an opponent, since these poisons could be delivered by a fairy-sized dagger/needle, though guns (or bows) would allow a fairy to deliver a toxin at range.

Oh, that's easy!

The size and speed of the bullet only matters when you're relying purely upon kinetic energy to take down the target.

However, if you have relatively tiny fairies who literally can't handle the recoil of a weapon with a decent kinetic energy, change the paradigm.

Instead of trying to make a gun capable of punching a lethal hole in what is essentially a kaiju to these fairies (a fairy weighs around 0.5kg, while an average human is 75kg, that's a 150:1 mass ratio in the humans' favour, way more than humans vs elephants), make a gun capable of delivering a lethal toxin to their enemies.

There are many neuromuscular-blocking drugs such as curare, or other toxic agents such as strychnine that are relatively rapidly lethal that could be placed in lethal quantities within a projectile that would be able to penetrate human skin, yet would otherwise not carry enough energy to be lethal except with a very well-placed shot.

So, as long as a fairy can get a poisoned shot from a gun into a human's flesh, biochemistry will do the rest fairly quickly. For a small being who is far better placed to make the best of available cover than a human, taking a shot and retreating until the toxin has done its job makes far more sense than standing their ground and trying to take down a human with what amounts to the fairy-sized equivalent of elephant guns.

So, with toxins, a fairy can be lethal with what amounts to a fairy-sized revolver or a light rifle. They wouldn't need to use any particularly heavy weapons against regular humans.

That presupposes that humans don't go after fairies while wearing body armour. However, when an armoured-up human comes along, the fairies can break out their equivalent of crew-served artillery and deploy airborne toxins... but that's an answer to another question.

Additionally, once it becomes known that fairies use poisons, they'll become feared just for that. A fairy wouldn't even need to carry a gun to be able to poison an opponent, since these poisons could be delivered by a fairy-sized dagger/needle, though guns (or bows) would allow a fairy to deliver a toxin at range.