That which is possible does not happen when good doctrine demands something else
Why portable man-guns will be too awful to be the best option
Can fairies invent recoilless small arms for killing humans? Probably. Will they use them, probably not.
The smallest guns capable of harming a human in the midwest were the tiny derangers women often carried which weighed about 1 lb and held a single shot. But these things rarely killed anyone since their muzzle velocities were so low that they basically hit like the .22cal pellet guns we used to shoot each other with as kids. They sting like hell to get hit by, but even basic clothing will usually stop them, and hitting exposed skin from more than a couple of feet away is likely to just leave a welt. More importantly, it takes a minimum of 2 pounds of force to set a spring strong enough to detonate a blast cap; so, a single fairy would need a winch or something to rearm the firing mechanism, even if they had extra ammo. Also, true recoilless weapons tend to only be half as powerful as their with recoil counterparts; so, you'd have to content with that as well.
So yes, they could make something, but it would not be good or effective as an infantry level weapon system.
Why doctrine will focus on bigger cannons instead
Instead let's consider this from a doctrine perspective: Do you want 4 troops with ineffective weapons and one bullet each, or a team of 4 fairies manning a rapid fire cannon than can reliably take a human down with each shot. The logical answer is that they will CHOOSE the cannon every time, even if the deranger is a possibility. Once you are thinking in terms of 4lb weapon systems, you can make something comparable to Colt P1840. These 45cal revolvers weighed about 2.5lb which you could break into 3 components that all weigh under 1 lb: the barrel, the cylinder, and you would replace the trigger and handle with a spring traversing carriage. Then your 4th team member could carry up to 28 rounds of ammo.
Based on the P1840, your carriage would need about 4 inches of spring recoil distance to absorb the shock using its own weight without flipping or sliding your cannon out of position.
What this means for your gun crew is that your 4 fairy team could fire at and kill humans from dozens of yards away, and they could fire 6 shots in rapid succession with 2 working together to pull the hammer mechanism, 1 to aim, and 1 to fire. And then to reload, they could all work together to quickly refill the cylinder. So, not only is this going to be more powerful, but it will be able to fire more shots over time.
Lastly, you need to consider how humans will fight fairies. Humans already had multi-shot shotguns at this point in time and many were seasoned bird hunters able to shoot down fairy sized targets at pretty long ranges. Or worse yet, humans had punt guns which could drop a whole squad of fairies with a single shot. This means that fairies will be much better off fighting humans on the ground where they can entrench, and their small size makes them harder to spot. This means they need to focus on cannons with longer ranges and well hidden entrenchments if they want to get close enough to fight humans.