Using a volt meter on the end of the 7 wire cable can be done pending what you are testing. If he was checking to see if the battery charging wire was live with power, that is an OK use of a volt meter on the 7 wire cable.
Not knowing what pin he probed on the end of the 7 wire cable, I'll make an assumption he knew which pin was the battery charge line and which was the DC common to do a volt check. And it sounds like this turned up no voltage if we understand your note correct.
1. A few questions, was the camper plugged into shore power when he was doing this test?
2. Was the battery disconnect with turned on or off when he did this check?
We need to know the answers to those 2 questions before we make any conclusions.
Inside the junction box on the front frame of the camper is a "battery charge" 30 amp fuse. It is to protect the camper and the truck from a dead short or over current on the charge line. When this fuse blows, it is hard to even know it is blown until you do some testing to find out why something is not working, like the camper battery is not being charged when you are towing down the road.
To test if the battery charge line is working, you can only test for power at the 7 wire plug when the battery disconnect is on. If it is off, then there will be no power to test from the battery. BUT, if the camper is plugged into shore power, that same wire is powered then by the converter in the camper. So the better way to test this is, make sure shore power is unplugged, the battery disconnect is on and then test.
You also do not need a volt meter to test this, you can use any camper light. If the truck charge line wire is working and the truck 7 wire plug at the back has power at it, (some trucks need the truck running to test this) then just plug the camper 7 wire cable into the truck. Have the battery disconnect switch "off" in this case, and then turn on a light inside the camper. It can be any light you know worked. Even the hitch light outside. If the fuse is good, the power will flow through the battery charge fuse to the light switched on in the camper from power supplied by the truck. If the light does not work, then the charge fuse is blown or has real bad corrosion on it.
This battery charge fuse is totally separate from your tail light issues. It is a problem and the fuse should get replaced so the camper battery is charged by the truck when you are towing. But it will not fix your tail light issues.
Someone will need to get into that plastic junction box on the outside frame of the camper behind the battery. Getting that cover off is real pain in the neck. Odds are high dirt has filled the screw holes and there are tiny Philips screws buried deep inside.
See this post for what it looks like inside and the charge fuse holder. While this post deals with fixing the 30 amp auto resetting circuit breaker, a the bottom of my reply it talks about how to get that cover off and not break it in the process. If you get the cover off, I suggest you also convert to the screws I have listed in this post so next time it will come off a lot easier. If you have the repair man go in that box, make sure you show him the pics in this linked post and how to get those screws out. That box is fragile and it will not take much to break it and then you have to deal with that.
Also while he is in there, this is the place the 7 wire cable splices into the camper wiring and where the running lights and the stop/turn/hazard lights wires start. He can check those connections for corrosion.
Since it now sounds like "all" running lights do not work, is that correct, "all" running lights? If so then there is a possible issue with the prongs on the end of the 7 wire cable. This can be tested for if they are actually working or bent and not making contact. This comes back to needing to do a good test on the end of the 7 wire cable to confirm what does and does not work.
Hope this helps. These electrical gremlins can be search and destroy mission to find some times. A lot of digging for loose or bad connection somewhere.