[DaveP.]

Fuses #1 & #3 underhood will only show voltage with the service brake applied or when a turn signal is on. I use Hazard Flashers for checking rear lighting. In a 7-pin RV connector, the Left and Right are the terminals at 9:00 and 3:00. These can also be checked with a meter while Hazards are on.
There are 3 more fuses in the underhood for "trailer tow": Battery charge (Maxi 30), Brake Controller (Maxi 30) and running / tail lights. I don't remember the fuse number off-hand, they are in the manual. The only circuit that needs the key in RUN to energize is the Battery Charge. The lighting circuits don't. (If the light on the rear of the trucki is on, the trailer plug circuit should be live too).

Here's a diagram of the 7-pin RV plug. While it is possible that a previous owner changed the wires on the terminals, it probably didn't happen. But easy to check.

 

[DaveP.]

So does this mean it's likely an issue with the trailer wiring, rather than the truck?

I'm not clear on what you had the meter leads on, and what was on in the truck, ie what the conditions were. BUT: If you are "getting power" on the other 7-pin in the truck (including fuse #1 & #3), the TRAILER could have a bad ground pin or connection. If the trailer isn't grounded back to the truck, this condition will force the various light circuits high (12 Volts) when the circuit is hot, but the light on the trailer won't be on. Because the two 7-pins are in parallel, the bad trailer ground will force both 7-pins high. That was a good test. Good thinking.

 

[DaveP.]

You mentioned that you have a "test light". Put the ground wire of the test light on a chassis ground point, like the frame, and be sure you have a good connection. Don't have the trailer or anything else in the other 7-pin that you're not testing. Using the diagram I posted for the pin positions, check each pin in each plug with the test light. Do only one test at a time, and check every pin. Ie turn on Left Turn. Put you test light on the pin at 9:00. The light should flash in unison with the truck's LR turn signal. Check the other pins, there should be no light on any of them except 12V Battery Charge (1:00 position) if the key is in RUN. Check each pin in this manner. Battery charge should be+12 Voltt with key in RUN, and 0-Volt with key in OFF. The BRAKE terminal can also be checked with a test light, but may or may not be full brightness.

For the trailer plug, you can check the ground by putting your test light between the Ground (7:00 looking into it, with the index tap at 12:00) and Battery Positive (11:00) The light should light if battery switch in the trailer is ON, the battery is charged a little, and the wiring is intact.

Those plug-in testers can give misleading results, especially with bad grounds, because they have LEDs and not incadesceant bulbs. Try the tests I suggest with the test light, and be sure both 7-pins on the truck are behaving the way they are supposed to. If not, report back with your findings, we'll go from there.