Another good reason to power an inverter from the batteries. The power from the converter may not be that clean, an AC carrier could be in the output and the inverter may pass that ripple through to a device. A battery acts like a big capacitor so any AC on the converter's output or a DC spike will be eliminated.
A typical surge protector on the AC from the inverter will probably not save your bacon, most of these are typically MOV devices that only drop connection when there is a lot of energy to dissipate (think dead short or lightning). A filter and surge protector may catch any nastiness before it hits the speaker's PS. The price of a different set of powered speakers or using passive speakers connected to an audio output may be less than rewiring or buying an AC filter.