The first 50 feet or so of the east side could definitely be considered 5th class climbing. It's easy, but I wouldn't want to downclimb it without a rope. Also the last 30-40 feet to the summit is super easy to climb up, but the huge exposure could make the downclimb seem unreasonable. There are sweet rappel anchors at the top and nylon webbing anchors scattered about the climb which make rappelling the easy and safe option for the descent.
I just came back from 3 days of climbing at Joshua Tree NP where running it out is the name of the game. I'm comfortable climbing high unroped and going long distances between protection. I would not want to downclimb Weaver's Needle.
Whatever you decide, have fun!
"I'm not a vegetarian because I love animals; I'm a vegetarian because I hate plants." A. Whitney Brown