1 the battery switch should be on to charge. This connects it to the trailer charger (converter)
2. This can be involved to track down. Generally poor wiring or shorted wiring. Moisture could contribute but only if it has a way into the
electrical system. Could be broken or cracked components. Others with a similar system will probably jump in here with more help
3. Your voltage readings. The 12.5 to 12.9 in full sun sounds OK. Some will go a little higher. However the no Sun readings as low as 10.8 may be a problem. These readings imply you have a standard lead acid battery. Not lithium. When there is no sun your voltage reading is just measuring the battery voltage. A lead acid battery will be damaged if drawn below 12 volts. 10 volts is a dead battery. It may temporarily have a reading below 12 volts under a load, such as the water pump turning on but should rise when it shuts off. Small loads such as a
light build will lower the reading a little but 10.8 is very low. Let the battery rest for a while without any load and not connected to shore power or your
solar. A full battery will be slightly above 12 volts and it should never
fall below 12 volts for maximum battery health and longevity.
4. If you have a good trickle charger, one they automatically switches to a maintenance mode when the battery is charged, that should be fine. Really a battery maintainer is what is needed for storage. These put out a smaller voltage and will not overcharge and damage the battery. Many but not all trickle chargers have this automatic function. You will have to check for that on the specific one you have.
Happy Halloween