Mr. Saif can you tell us that are u using half bridge inverter or full bridge inverter?.
If you are using half bridge, better go for full bridge version. There by the battery voltage requirement comes by Half.

Increase the secondary voltage rating of the transformer to around 250 volts, upon loading the voltage coming down will not have any effect on your domestic loads.
Voltage drop you are seeing is because of the drop in transformer, due to increase in impedance. This is caused by square wave input to it.
second reason there may be reduction in terminal voltage of the battery because of loading also.
One more sujjestion is before sampling, feed the square wave (Quasi-suare/sine wave) to precision rectifier( OP-AMP Based) and fed to the ADC.

Better go for PWM mode control than simple square wave..
Gowrish