The solar controller will only put out power when it thinks the trailer thinks it needs it. So yes, the low output is because the battery is full and the shore power connection is keeping the voltage up.
The "unused" power from the solar panels doesn't really go anywhere. The panels will be at their full voltage, but the solar controller is refusing to take more than a tiny amount of the current that is potentially available. In (simple) electrical systems, power equals voltage times current, so when current is near zero, there isn't much power. In your case, the power is just not being created even though there is potential to create that power.