Hi
First off, voltage readings are only a *guess* at the state of a lead acid battery. As others have noted, there is a wait time. They also are highly temperature dependent. If there's snow flakes blowing around 13.1 may be a "half full / stop using" battery.
Next up, the voltage readings in a typical trailer are taken well away from the battery. Put on a large load and you will get a voltage drop. ( If the same path is used for the load and sensor). The battery is still fine, it's just the load throwing off the reading.
Indeed, there *could* be a very real problem. There's no way to really know with a single reading. Grab a digital multimeter and measure at the battery posts. That will take the whole "load on the wiring harness" out of the equation. An even better answer would be to install a shunt based battery monitor.
There's a wide range of things that *could* be going on. You could have a loose clamp on one or the other battery post. The batteries might not be charged (stock charger will not fully charge them at freezing ....). The batteries might be shot. There might be a loose wire someplace else in the wiring harness. Everything could be fine. The only way to sort it out is to dive a bit deeper.
Best bet - get a Victron BMV-712 ....
https://shop.pkys.com/victron-energy...y-monitor.html
Bob