Is there a voltage inverter IC, 3.3V to -3.3V with >1A output current?
If not, any idea of a circuit to do the job?
1A output is not so small that you could really end up using significant size diode at output. You may also take a look at ADP5074 which is really an inverter but per my checking can only deliver up to 0.8A for your requirement voltages. The switch is internal so you only have diode and inductor external components plus capacitors of course.