How do you feel about the prospect of busking chained to a cold water pipe? That might solve your issue.
What type of pickup are you using?
An experiment that might shed some light: with your amp plugged into the inverter running off battery (and you hearing the noise), plug some other input or output of your amp into some other piece of equipment that is grounded to earth with a 3-prong AC mains connection. If the noise does not significantly reduce or go away altogether, then the problem is most likely that your inverter and your amp are not playing together well.