Well, I don't have a recommendation for a break action barrel pellet rifle.
But I can recommend a Daisy 953, an underlever cocker. I upgraded mine with a Daisy competition rear sight, and a pyramidair globe front sight. I ordered the sling and sling mount directly from the Daisy customer service department.
Used to be you could get the Daisy Avanti 853 from the CMP (from ROTC shooting programs). The 853 is the more expensive version because it uses a Lothar Walther barrel and laminate wood stock. So far I haven't been able to shoot the 953 accurately enough to deserve an upgrade, so I will keep training on it until I wear it out or get good enough to need a better rifle.
I do my training at 33 feet, aka 10 meters, shooting ISSF bullseye targets. So far I have managed to get more than half the points on an 8 bull page only once.
Machine guns are awesome until you have to carry one.