2 things to check.
When were the batteries last "fully" charged and by what means ? How do you know they were fully charged ?
Second, a 1000W inverter could, in theory, draw over 80 amps ! Highly unlikely if the refrigerator was the only thing running. What are the specs for current draw from the refrigerator ? An "average" 18 cu ft residential refrigerator drams about 6-8A @120VAC or roughly 700-800 watts, when it is running.
It really depends how many minutes per hour it runs. If you shop around, there are some refrigerators that take less.
You really should invest in a AC-DC clamp on ammeter. There are many out there to choose from that are less than $100. I have this one
this one.