One ought to be able to prove this for most braid groups in a similar way to $B_3$. It was shown by Venkataramana that Burau representations of braid groups are arithmetic in the appropriate range. Arithmetic groups should have lots of congruence quotients which are perfect by the strong approximation theorem. But I don’t have quite enough knowledge of the appropriate group theory to complete this line of argument.
This should follow from the approach to prove Theorem 1.2 of Masbaum-Reid. (Alan Reid suggested this to me)