Let's make this simple. There is only so much 12volts running back to your trailer from your alternator. You can either charge your
battery so that when you stop it will be fully charged, OR, you can operate a 110 refer running thru an inverter. I doubt that you can do both. As the other Norm suggested, the refer will stay cold for 4 hours. So, you can then start operating it again after 4 hours and by then your
battery should be charged from driving for 4 hours. But, if you turn it on from the beginning, you will have a cold refer but a discharged
battery.
That is why it is recommended you purchase a 12v refer. They draw less amperage and you can operate it and charge from your battery at the same time, unlike above. There are a lot of choices one can make. Some are simple other are more complex.