A useful general strategy to tackle such questions is to use small cancellation theory. For instance, in Small cancellation in acylindrically hyperbolic groups, Michael Hull proved the following statement (known before for hyperbolic and relatively hyperbolic groups):
Theorem. Two finitely generated acylindrically hyperbolic groups admit a common acylindrically hyperbolic quotient.
As a consequence, because there exist many perfect finitely generated acylindrically hyperbolic groups and that quotients of perfect groups are perfect, every finitely generated acylindrically hyperbolic group admits "many" perfect quotients. This applies in particular to braid groups: even though $B_n$ is not acylindrically hyperbolic, so is its quotient $B_n/Z(B_n)$ for $n \geq 3$.