Most of my suburban pest removal the last few years has been with a .177 at 850-900fps (14-16fpe). That gives a nice flat trajectory for the 50-60 yard shots that arise from time to time, however the vast majority of my shots are in the range of 25 - 35 yards.
Just in the last few months I began playing with low power tunes--mostly regulated PCPs--for back yard fun shooting. By low power I mean 550-650fps (6-9fpe in .177). After building confidence in how well they group, I decided to try them first on the smallest of pests like rats, chipmunks, and house sparrows. The result? A resounding success.
Then I decided to step up to gray squirrels. Brain shots only. Anchors them where they stand. Even wadcutters have no trouble penetrating their heads. I have captured some on scope cam...haven't processed most of it but here's a short clip (1 minute) showing a couple of chipmunks
https://www.youtube.com/watch?v=TDxLiw86DZ0
So far, I've only experimented with slow .177. I haven't tried slow .22 or .25 on larger critters. The slowest I've gone on raccoons is a .22 at 750fps (20fpe) and those were complete lights out with brain shots.
FWIW, I've had no trouble finding pellets that will group 1-2 MoA at these distances and relatively low muzzle velocities.
Now the caveats:
1. I don't have to deal with much wind where I am, and it is true that domed pellets typically achieve their best ballistic coefficient in the 800-900fps range. In windy conditions, I would think twice about picking up a 550fps airgun to shoot at a living thing.
2. Ranging is critical. For example on my 560fps .177 shooting RWS Supermags, from 20 yards to 35 yards, there is an almost 3 inch elevation difference. If I get the distance wrong and use the wrong holdover, it's a miss. Or worse, a wounded animal. That won't do. We're talking about the back yard here so there's no reason to get the distance wrong. Just have a good range card and use it.