Not many of the cheaper ones would have an auto change over, The larger Victron can be selected for Inverter or battery charger mode, I would if on power select battery charger, it also puts power through to power points fridge and ho****er system. If free camping only switch on to inverter mode when actually require 240 volts.
The seriously expensive inverters are double conversion. They are running off the batteries all the time while simultaneously charging, for critical network computers etc.
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My Renogy inverter is only used to power the fridge which is 230v only. The inverter is always connected to the battery but also has a 230v connection to the van, this power is passed through directly when available, no battery drain. As soon as 230v power is lost the inverter auto changes to battery and maintains the supply to the fridge. While driving the tug maintains the battery assisted by solar when sun is shining. Nothing to it, no switches to remember.
I am only using the inverter to power the fridge when mains power is not available, it is not connected to the rest of the van. Larger inverters are available that could power the whole van in a similar manner as long as adequate battery was fitted.