You can camp for a few hours during the climb but not for the entire night. You have to reach the trail marker, with your entire party, before the timer runs out. That usually leaves several hours for doing other things, including sleeping. In my experience, each event only ends when the timer runs out, you end it early by clicking on the timer button, or you move on by clicking on the Move On button.
My current Sim bought her own tent, grill, cooler, and campfire, and is carrying all of them in her personal inventory. The tent is from Snowy Escape, the grill is from Jungle Adventure, and the cooler and campfire are from Outdoor Retreat. That gives her three ways to get food and she has actually used the grill. She hasn't used her own tent yet but she did sleep in the tent at the base camp, without ending the event. She also took a Disco Nap in the bush, on a different climb, and that didn't end the event either. The key is that her sleep/nap only lasted for an hour or two, so she still had plenty of time to reach the trail marker.
Obviously, the tent is for sleeping. The grill lets her make full meals by grilling. The cooler should let her get quick meals when she doesn't have time to cook. The campfire should allow her to roast marshmallows, hot dogs, and various harvestables, including potatoes, which I found growing in the second area. Unfortunately, my Sim's rock climbing skill wasn't high enough for her to get past that point, so her climb ended when the timer ran out.