If $n$ is even, then the action is not locally ballanced. You can choose $B$ to be the conjugacy class of an element of $SO(n)$, as close as you like to the identity, that does not have $1$ as an eigenvalue. Now whatever $x$ is, $x$ will not belong to $Bx$.
On the other hand, if $n$ is odd then every element of $SO(n)$ does have $1$ as an eigenvalue, so if $B$ is any conjugacy class then $x\in Bx$. Furthermore, $Bx$ is preserved by all elements of $SO(n)$ that fix $x$, and is connected, and (if $n>1$) does not consist of $x$ alone; and this makes it a closed ball centered at $x$. Therefore for general balanced nonempty $B$ the set $Bx$ is a union of such balls.