I am not sure just how long the power line is from the alternator to trailer. Those long lines have a lot of drop, and even when running shore power those fridges can have problems.
If you can run on gas, I would. That was not possible on our old Trailmanor we had. We would have to hookup to shore power or the genset to refresh our batts at the end of the day.
Part of the reason why our TM had that problem was because the battery was located in the rear of the trailer. That made those lines even longer. If your Chalet has the batts on the tongue that may be enough.
The additional problem you have is the small size of your alternator. 60 amps just isn't that big. Also the 7-way plug can develop corrosion on it, especially the power connection.