The Xantrex ProWatt SW 2000 does not have a floating neutral.
https://forestriverinc.com/files/Com...;s%20Guide.pdf
The ProWatt housing is grounded. The 120 volt dupex outlet has ground pins. The output ground pins should have continuity to the RV frame ground. Any ground cabling from inverter to remote outlets should have continuity to ground.
The ProWatt does not have a transfer switch, so the ground connection is not switched.
Test the duplex outlet on the ProWatt using the outlet tester. If there is proper ground there, then the open ground is in the extension cord from the inverter to the refrig outlet.
Outlet testers come with many different capabilities. Sometimes the simple logic circuits become confused and present strange results.
I expect there is an open ground in the circuit. But to be sure I would get out my cheap digital multi meter and measure voltage to frame from outlet ground. It should be zero volts. Then switch to ohms. Ohm reading should be near zero to verify outlet tester.