Main Advantage
Their main advantage would be that they can start out with the ability to destroy dirt / build tunnels without dynamite (though dynamite will be much faster later in the game), so right away from the start they would dig underground into their cave home. Maybe there could even be a second step where they can dig faster by employing diggers (they need pickaxes or shovels made from wood / metal, whatever is more balanced)

Reproduction:
Their "houses" (maybe more like bunk beds or something) have to be built underground. Maybe it is even more about "underground space" which limits how many beavers you have, i.e. every beaver needs a certain amount (2-3 blocks?) of "underground space" to sleep, so the larger you make your underground area the more beavers you can support. Since detection of this might be difficult, maybe adding selectable "underground home space" like you do crops/trees/harvest space would be good, then beavers would just put down sleeping bags in that area or something. This would be cool since it is a different mechanic to both Folktails (discrete houses) and Ironteeth (essentially a production building) for reproduction, being more like crops i.e. a selected area.

Special Behaviour:
Like the comment before, I also had the idea of making them light sensitive. But my idea was, they work slower in the light, until they get some mid game ressource (like a hat, maybe made from straw which would be one of their crops; or sunscreen from some oil, again from crops). With this it would also make sense to have their day shifted, i.e. they work at night, but sleep at noon.

Food:
For Food I also came up with mushrooms, which have to be grown underground (so they can easily do that before mid game when the light issues are dealt with).

Energy:
I see two ideas for their special energy generation method.
Playing on the idea of light sensitivity, maybe they realised the power of the sun, so they can build some sort of solar engine (solar panels would be difficult to justify why they produce rotation rather than electricity I guess). This would be kind of fun, since it produces energy the most when they sleep. Balancing could be how cloudy it is at any given moment.
The second idea would be a sort of biofuel generator, maybe from a second type of mushroom or some other underground specific ressource.

Movement:
Since both Folktails and Ironteeth have their own new movement option with ziplines and tubes, the movement option for the underground beavers could actually be minecarts. They build tracks and minecart stations and can then travel between them. Might be more limited since it is kind of like roads, having to built them on the ground, but they might be the fastest or (if possible to implement) they could also carry items from one station to another i.e. beavers can carry more with them somehow as a balancing point.

Other Ideas:
They should probably be really good at mining metal, so they would have the most advanced mine. Maybe they can even skip past scrap metal piles and instead go for a simple mine right away or at least sooner.

Maybe some building requires both underground AND water. That would be interesting to achieve and probably risky with flooding your underground area as a potential problem. One idea could be a "mineral grotto" which is one of their wellbeing buildings. Kind of like bathing but it has to be underground. But I don't know if just doing it for wellbeing would be worth it.