Should that occur, terminate the equalization process by unplugging from shore power. Also, the selection of AC appliances is much broader than in DC appliances, which is one reason demand for inverters in RVs is growing. In selecting a unit, be certain that the specs are accurate. Reliability and warranty are important. A two year warranty is a good indication of product confidence.
Safety is another consideration.
There are some things we can do to maximize inverter systems. For example, we can take steps to ensure the batteries are being charged properly. It's important to understand battery capacity and cycle life. Avoid excessive discharges. Precision amp hour meters can be extremely helpful in accomplishing this.