Short answer, yes. Running the fridge on gas while traveling down the road is a highly debated topic here. There are some restrictions on this (tunnels, ferries, when refueling, etc) but personally, I've been doing this for 7 years with no problems. Your truck battery should supply enough of charge to your trailer battery to run the fridge all day... If it doesn't, there's an issue with your charging system or that charger isn't making it to your trailer.
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2018 28BHBE
2017 Ford F250 XLT, 6.2 gasser
2013 26BH (traded)
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