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The first thing to check is the DC input to the fridge. Be sure that the batteries are fully charged and that the converter is putting out the needed voltage in order to charge them. Then you can be sure that isn't the issue. Most of the time, this will be either the light itself or the ignition module. Checking that with either a small digital multimeter or 12 VDC test light can tell you that. I can direct you in the testing. The meter is about $20 or so at most hardware stores and about $5 or so for a test light. Get back to me when you are ready to test. All follow ups are included in our fee and fully expected in these situations. I will make a phone call option available to you in case you think it might be helpful.
I am adding a link to the Dometic Service Manual which can be very useful here. Please get back to me if we need to dig deeper into the problem. Thanks! http://bryantrv.com/docs2/docs/rm2510_52.pdf