Mechanical radiation, aka "sound", is a more likely culprit. Inverters now use a technology called PWM (Pulse Width Modulation) in their design, what is known as a Switch Mode Power Supply (SMPS). Actually, the SMPS concept is used now in almost all electronics, it is what allowed designers to make things even smaller again after transistors came into use instead of tubes. In an SMPS, transistors are fired at very high rates, as in 1-15kHZ, and various combinations of on and off times, fired into inductors, allows the transformation of power with very little space needed or heat generated. But in that process, the transistor firing has an effect of making the inductor components vibrate mechanically, which produces a high pitched sound. Remember the hearing tests you were given as a child, where you put on headphones and told the nurse or technician if you did or did not hear a tone? Those tones are created by PWM, and are the same thing that comes off of all SMPS based devices. So the PC you are using has an SMPS, the monitor has an SMPS, the phone charger has an SMPS, the TV in the room has an SMPS, LED and CFL lamps are all using SMPS inside, all of them emmitting high frequency sound. If you have an inverter (whatever that might be), it too will be an SMPS. It is entirely possible that the multiple SMPS devices in your environment are emitting complex sound patterns that, even if you don't "hear" them, are nevertheless having an effect on your auditory nerves. Adding the inverter might just have been the final straw in a big pile of hay.
Simple answer, try moving everything around, envisioning each of them as a sound generator, so you can locate them in ways that might disrupt any resonant frequency interactions. I used to get migraines so I had to do that, it worked wonders. The more powerful the device, the more powerful the sound pressure coming from it, so I started by putting my UPS under the desk, that made the most difference. I then moved most of the "wall wart" power supplies for various things to one of those "squid" power taps and put them behind my desk by pulling it away from the wall a little, then filling the gap with a board. The only SMPS devices on my desk now are the laptop itself and my desk lamp, because it is LED and has a USB charger port built in. My migraines have never returned.
By the way, it will have nothing to do with batteries, other that to keep the batteries charged, the charger will be..... an SMPS!