could you post a diagram of how the four gate signals should look like
Here's a waveform picture from a power electronics lecture showing unipolar (also called 3-level) pwm
Uan and Ubn are the outputs of both half bridge legs, expect respective high and low side gate waveforms (high side similar, low side inverted, some dead-time added). U2 is the resulting bridge output voltage. The first line shows the respective triangle voltage of an analog pulse width modulator.
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Regarding switching frequency discussion, 16 kHz is the minimal value if you want to avoid audible noise, sounds basically reasonable.
The interesting question is how much core losses are caused by the pwm frequent current. High frequent AC flux is only a small fraction of 50 Hz flux, the additional core losses are probably low. Another point to consider is skin effect loss of the most likely massive primary wire.
You can of course add a dedicated LC low pass filter before the transformer, but most designer would try to avoid it.
The preferred todays topology for kVA battery inverters is high frequent DC/DC + high voltage inverter in any case.