You don't need chains for the trailer. They are for the drive wheels of the tow vehicle. You might consider some protection for the front of your camper if you will be in an area where they use a lot of sand/salt on the road. Also, if you don't have a cover for your propane
tanks you could cover them, especially the valve assembly and regulator, with a plastic bag or other protection.
You should also have winter survival supplies with you in case mother nature decides not to follow the weatherman's suggestions (i.e. forecast). All the things they recommend for winter travel such as blankets, candles, food and water should be available. This brings up another point. Take along a weather radio so you can listen to the NWS weather guessers.
Best suggestion - slow down. Slow way down. Use the manual control on your trailer break controller (lightly) to maintain control if things get a little hairy. If you notice any changes in how the rig is handling slow down some more. Leave lots of room around you. And if others on the road start crowding you just back off and let them ride point. Let them find the slippery areas for you. Patience and caution will keep you alive.
Depending on where you are traveling you might have to "wait it out" if the weather gets really bad. In some areas they prohibit trailer combinations on the highways or close the roads when the weather gets bad so you might end up sitting at a truck stop or roadside rest area with a bunch of BIG trucks. I'm talking about really bad weather here, like snowfall measured in feet and/or blizzard conditions. BTW those big trucks make good windbreaks. As long as you have your camper you have a place to stay. If you don't have a furnace
keep a small single mantle lantern for heat, and remember to open the camper vent if you use it.
I just completed a 2,400 mile trip during the holidays Denver CO to Danville IL and back. Some on interstates and some on state and county roads. Even a few miles on gravel roads. Had snow, ice, wind and torrential rains. Low temperature was 3F with wind chill of -16. When the ice storm hit we just pulled in to a rest area and dropped anchor until it quit. Overall it was an enjoyable trip.
Best of luck on your trip. Be ye cautious!