5A 12V Battery charger -> 12V Marine Battery -> Inverter -> Fridge for longer run-time?
Maybe, though it's likely to be a very inefficient use of the propane.  You can get fridges that run directly off propane.
If it's a smart battery charger, then it may be confused by the varying load on the battery.  This will cause the battery voltage to go up and down unexpectedly.  Smart chargers work out where they are in the charging cycle by monitoring the battery voltage.
The charger is only putting out 60W maximum.  Is that going to be enough to do anything useful?  A 1000W inverter is likely to draw a current, even when the fridge thermostat is off.