Depending on the vintage of your fridge, it may have a manual pilot light, or more likely an electronic spark ignition. The "clicking" may be the ignition spark attempting to light the burner.
I would start with your battery connections. The clamps should be tight and free of corrosion. You may be able to follow the ground cable from the battery to the frame or where it ties into the DC distribution connections. Check that they are all secure. Also find where the 12V
and ground connect to the back of the fridge and check those connections for tightness. While looking at the battery, add distilled water if needed to bring the electrolyte level up to within 1/4" of the bottom of the fill holes. Be careful about splashing the electrolyte as the sulphuric acid is nasty stuff. It will burn holes in your clothes and worse on your skin or eyes.
I would then use a voltmeter to measure the voltage during the day when the fridge is working correctly, and then again at night when it's not to see if you do in fact have a low voltage condition when the lights are turned on.
Someone else may know where the converter is located in your trailer. It is typically under a bed or seat. Older ones are known to hum, so maybe you can find it by listening. A converter that is operating should put out about 13.2 to 14.5 volts give or take depending on the charge level of the battery. When the converter is off, a fully charged battery should read about 12.7 volts. A battery that reads 12.0 volts is considered discharged for practical reasons. If the voltage gets lower than that, circuit boards and gas valves my not function as intended.
When tightening any electrical connections, be sure your shore power is disconnected as well as the (-) post on the battery to prevent any arc welding should your wrench or screwdriver get across unintended points.