If your battery dies in 2-3 weeks you need to find the battery disconnect, that will protect the battery while in storage. There are a number of things in the trailer (propane/CO detector for example) that draw power all the time unless the disconnect is pulled.
I don't keep mine plugged in all the time because it seems to put more load on the converter, the fan runs frequently. I plug it in about 5 days before I am going to use it to make sure the batteries are charged and to bring the fridge online for packing. I also charge the batteries every couple of months if it's sitting for that long using a good quality charger.
I would recommend looking into a "battery maintenance" style charger for your converter. I was able to add one to my converter for about $25. It allows for float charge as well as a discharge cycle (discharge for 1 day, then charge). This allows me to keep the trailer plugged in without overworking the battery, it has also cut down on the amount of water loss in the batteries. The larger Battery Tenders will work as well.
If you are leaving the trailer "powered up" on 12v make sure that any maintenance charger has enough power to keep everything going. Some of the small ones are only 1-1.5 amp which may not be enough to keep all the 12v devices running.