The Belters have always been an oddity, and something the universe never focused on. They are known to have high technical proficiency, and zero interest in the affairs of the Inner Sphere. Although the reference eludes me, it was one of the Jihad books or maybe the IP series, even the WoB knew next to nothing about the Belters and decided not to expend the effort required on what would likely be a failed effort to bring them to heel.
The belters builot a fairly large proportion of Sol's jumpships (in fact, in JHS terra, it's called out as ironically one of the things that led to their temporary decline), and they actively need them-- a civilization spanning the asteroid belt to the oort cloud isn't going to be functional with STL transports.
As a society, with 1.5 billion people who didn't have any technological regression from the wars, or any effective suppression by comstar, the fact is that the Belters should be not simply as advanced as the old star league, but probably more advanced than the clans, at least in most of the theoretical sciences (not having any need for heavy ground weapons, it's likely that they didn't convert those developments into advanced tech). Hell, if there's any place CASPER tech could survive, you'd find it in the belt, given how many benefits low AI tech would give them.
Then there's the problem that they can go anywhere. They have FTL, they have no need for habitable worlds, and so, they can literally live anywhere they have mass and energy--including things like brown dwarfs and other steller bodies tht the rest of the Inner Sphere would never look for.
BUT, in order to avoid "The mighty belt empire rules the milky way" they nave a few factors militating against expansion.
1. They don't need it. Being able to exploit the asteroid belt and other factors, coupled with making your own living space, means that they have no reason to expand--Sol gives them everything they need with room to grow into the trillions.
2. Long lives+an environment that requires technologically skilled citizens= a low birth rate. Belters don't have to have ten kids to work the fields. Children are valued an invested in, and that means that their rate of growth is very slow. Coupled with 1, why spread out? To go to the tropical paradise world? That's available in the Oort cloud, a habitat with steamy jungles and "safari" expeditions, all for the price of a cheap jump ticket. Their may be, and probably are other belter colonies, but they're likely small, with the same slow growth you see in the belt, and mainly settled by the kind of non-conformists that don't want to talk to other people.
3. Finally, the belters have a relatively sane culture, one that has managed to avoid the general love of wars over an empty throne. TGhey probably assume that any contact with the Inner Sphere or clans is likely to see annoying people show up and demand they use their higher technology in order to build stuff--and they already had that ride with Amaris.
OUT OF GAME... Well, the belters are recent addition to Btech in terms of the history of the game, and so you really have to come up with reasons why an advanced, tehcnological culture, without Comstars attitudes, didn't have a bigger impact onthe setting, when you consider that it's entirely likely that the belt can still, if they want to, turn out warships.