Since most modern RVs don't "HAVE" to have a battery installed, one can assume a properly functioning converter will provide all the 12VDC needed to operate all the 12VDC equipment while plugged into 120VAC shore power.
re: battery disconnect ... if it's wired normally, then OFF and ON indicate if the battery is disconnected or isolated (OFF) or connected (ON).
Assuming your cutoff switch is at the battery, there should be a red cable from battery to switch, then another red from switch to junction block or fuse.
To troubleshoot, following the electrical path from battery to fuse panel - note below that RED cabling may be or may change to BLACK somewhere in the path, as the system changes from automotive to RV standards.
While unplugged from 120V AC (to eliminate DC feed from converter), meter between the POS and NEG posts on the battery. That's your battery voltage (and may indicate higher than 12.5 due to surface charge from the converter).
Meter between NEG post on the battery and cut-off switch at the red cable from battery to switch. You should see the same voltage as at the battery terminals. If yes, then this cable is good.
Meter between NEG post on the battery and red cable on the other side of the switch (going from switch to fuse or junction block). If the switch is on the OFF position, you should read 0 VDC. Turn switch ON and you should read the same as at the battery.
Meter between the NEG battery post and the other end of the red cable at the junction box or fuse. Again, with switch ON, should read the same as the battery.
Finally, meter between the NEG battery post and the other post on the junction block or fuse. Again, should read the same as the battery.
Going inside to your breaker and fuse panel, you should have a lug marked DC Positive + or similar. While still disconnected from 120 VAC, meter between that block and the block containing all the white wires, perhaps marked DC Negative - or similar. Again - should be very close to battery voltage (minus a little bit for voltage drop.
Meter to the output side of each fuse and again should read close to battery voltage. Any that show 0 VDC indicate a blown fuse.
You can verify by pulling the fuse, inspecting and metering for continuity.
I hope this doesn't sound condescending ... if so, please forgive me. Here's a typical 12VDC wiring diagram (simplistic but representative).