You can use a multi-meter to see what your converter is outputting at the terminals.
As for replacing batteries; Whenever I hook up or disconnect the batteries I make sure that the 12 volt
master disconnect inside the trailer is OFF. I take extra special care to make sure the battery wires aren't touching anything that may cause an arc, spark, and short. When hooking up a battery I always attach the positive terminal first and then the negative. Reverse that when disconnecting a battery; remove the negative wire and then the positive. This will keep the arcing and sparking down to a minimum.
Hopefully somebody else can give you a lead on your two non-functioning lights. I am assuming you have already checked to make sure the bulbs aren't burnt out and they are making good contact with the socket. The multi-meter would also come in handy troubleshooting to see if you are getting 12 volts at the non-functioning lights.
Good luck and keep us posted.