Is there any reason for not finding an inverter that only converts from DC to AC maintaining the same output voltage, e.g. a 220V DC to 220V AC inverter?
Figure 1. A variable frequency drive schematic. Source: VFDS.
An inverter is similar to a variable frequency drive. The AC is generated by switching the DC bus voltage using pulse-width modulation in a fashion that generates a pseudo-sinewave. To generate a specific AC voltage the DC bus must be at least as high a voltage as the peak of the AC output required. For your 220 V AC output that is \$ 220 \sqrt 2 = 311 \ \text V \$. With a 220 V battery the best you could do would be 155 V AC.
To get 220 V AC you would need a step-up transformer after your 150 V inverter.