To add a 12VDC system is no small task. You will have to run a totally different set of wires for the new 12VDC lights. Also, you will have to consider some way to charge the 12V battery. The easiest way to do this is from the tow vehicle. If you have a 7 pin, (six on the outside and one in the middle) trailer connector, one of them is 12V power.
You may also consider a power converter to convert 120VAC to 12VDC. There are versions that will also act as a battery charger.
The fact that some of your 120VAC lights work and some don't tells me that you probably have a wiring issue. Copper corrodes. Where it corrodes it causes a high resistance point on the circuit. When current flows through these high resistance points, heat is generated, and wires may break. At 120VAC this can be quite a hazard.
Were I you, I would probably convert all the 120VAC lights to 12VDC LED lights
. I would pull out the 120VAC wiring. Before I pull it, I would tie some #14 seven strand wire to it and use the 120V wire as a fish tape to pull the new wires into the same place. Then using a converter/battery charger, I would power all the lights off the converter/battery.
You will probably have some 120VAC plugs in the trailer as well. These can be useful. I would examine the wires, especially the ends where they terminate in the outlets. If they are green with corrosion, I would replace them. Standard #14 solid house wiring will do. These will connect to where the 120VAC comes into your trailer, preferably on the other side of a breaker.
That brings us to the subject of over current protection. Every wire should be protected by either a fuse, or a breaker which is rated for the size of wire you are using, 15 amps for #14. The DC Circuits will need an over current device as well.
If you are getting the impression that this is not simple, you are correct. given that you asked if a 12V battery can be used with the 120V system, I am assuming that this is new to you. You should probably consult an electrician, or a qualified RV service shop for advice.