This will happen if the lights are not getting the full 12 volts. Which is often a ground problem. LEDs will work as you described, some lights illuminate a little bit, maybe flicker, and others, like brake, will do nothing.
A multi-meter will only tell you voltage, so checking all connections will not tell you if it is getting a good flow of that 12 volts, which is also called amps.
Check from the vehicle that you get power to all the trailer plug connections.
Check that your trailer plug is not corroded. The 7-pin plugs get a lot of green "rust-looking" mess on it. scrape that off. Use vinegar to help. Disconnect from the camper battery, then dip it in the vinegar, let sit, then use toothbrush or something to clean it.
I prevent this corrosion to a large extent by using a circuit breaker so I can disconnect the 12V positive from the camper battery to the 7-pin plug. This way there is no electrical potential in the moist air around the plug. a 12v positive so close to a 12v negative.
