I'd strongly urge anyone thinking of hunting/pest control with any kind of air pistol to engage their conscience and abandon the idea quickly.
As previous responses have highlighted the power and accuracy issues mean no shooter, no matter how talented and no matter the conditions, can be sufficiently confident of humanely dispatching the intended quarry in real world applications with a pistol.
Your opinion that no shooter can be confident of humanely dispatching quarry with a pistol is wrong. I have in the past dealt with infestations of rats and feral pigeons in enclosed indoor spaces such as lofts and cellars using first BSA Scorpion springer and later PP700 PCP pistols. In such conditions use of a 12 ft lbs air rifle would have been potentially damaging and possibly dangerous. An example of this was when shooting ferals in a large barn with corrugated steel walls. Using my 11.7 ft lbs TX 200 at ranges of 15 to 30 yards a loud clang was a sure sign of a clean head shot as the pellet passed clean through the head and hit the wall, imagine the potential damage or danger from ricochets in a 25x25 foot cellar or loft space.
I've had a number of instances where for various reasons my rifle shot on a rat has missed its mark by a few millimeters, but in every case the power behind the pellet combined with the fact you're not missing your intended POI by much has meant they've all fortunately down quickly.
Many of the rat shooting videos I've seen on you tube have featured some very dubious "clean kills" regardless of a rifle being used
11-12 fpe does more than extend your range over a pistol, it increases the trauma at the receiving end which can often be what is actually making the shot leathal if it isn't placed perfectly.
An important factor to be borne in mind is the terminal energy required to humanely dispatch quarry. For example a .22 rifle shooting 16gr ammo at around 543fps, equates to 10.5ft-lb. The pellets will shed energy at about 10% per 10 yards, and at 30 yards will be retaining around 60% to 70% of the original muzzle velocity, there would be in the region of 6.5ft-lb of retained energy when the pellet strikes home, so a pellet fired from a 5.7 ft lbs pistol will still retain around 5 ft lbs energy at 10 yards. the energy required to produce a clean kill in the case of a rabbit or grey squirrel, is around 4 ft lbs, the energy required to produce a clean kill on a rat or feral at that range somewhat less.
If the shot isn't placed perfectly I'd suggest more practice or more patience is required.
5fpe is the same amount of energy as a standard house brick imparts impacting at 3mph (and that's at the muzzle, the energy is dropping constantly from there on over distance). If you imagine the difference between the corner of a falling house brick landing on your big toe at 3mph and 6mph you'll have a good idea of the difference between a 2240 and say a 2250XL. 8mph and your feeling what a properly tuned rifle will deliver. Power matters, even with rodents. Leathal shots from lower power guns are possible but they depend greatly on shot placement. Not easy when your target is fast, twitchy and hiding in the shadows.
I would certainly not take a shot at a rat which was fast, (still moving?) Twitchy or one which I could not see clearly. Waiting for a clear shot may not result in a big bag of vermin but does the job cleanly.
Your comparison with house bricks landing on toes at different speeds is bizarre at best.
With a pistol it's pot luck where your shot lands in real world conditions, and as such it's a gamble as to whether of not it has the ability and energy to do the job properly when it gets there. Gambling and luck have no place in hunting/pest control.
No, it isn't pot luck, it is down to patience and lots of practice. I know from experience that at short range a pistol producing close to 6 ft lbs fired by a shooter who has put in the time to use it accurately has both the ability and energy to do the job properly.
Pistols which produce at least 5+ft lbs can IMHO have a role to play but only within a very particular set of circumstances.
The 22** guns are great fun, I love them, but the 2240 is a point blank dispatcher only, marketing them as anything more is irresponsible. The 2250 (and similar) can be passable for close range ratting but not much more. With a fair bit of tweaking and a lot of practice in varied conditions you can push the range out a bit but in my opinion any serious work is the domain of 11fpe+ from air.
My 2250 produces 10+ ft lbs of muzzle energy, you suggest that this is not enough and that only 11+ will do the job. I suggest you read through some of the posts on here regarding spring piston rifles being used by members which have been detuned, short stroked etc to 10+ ft lbs for smoothness and improved accuracy. As always accuracy not power is the main consideration and at 10 yards or less I would put a PCP pistol up against any spring piston rifle out there.
CO2 is inferior to both spring and PCP in consistency of power output and so it's not the best plan when buying a gun for live quarry.
Pistols aren't a plan at all. You're better off spending that money on traps/bait/a proper gun.
Used with an understanding of the characteristics of the power source co2 is a perfectly viable option, just because you don't understand it doesn't mean it is wrong.
With regard to advertising I don't think any manufacturer, whatever the gun or power source, should openly advocate the use of their products for pest control but down the years many manufacturers have used it as a sales tactic