Light on Yoga by Iyengar has some great tips for this posture in his book that may be helpful. Also you could explore drawing your legs in toward the mid line of your body thus balancing out your pelvis. Once your pelvis is balance there will be more space for your shoulder to move as it will not be pulled back as far. This should give you more space to level your clavicles and open your upper chest. Practice this pelvic opening and strengthening on the floor first where you are fully supported. The leg that is in front and bent, draw your toes back toward your shin and isometrically pull your leg back towards your body and in towards your mid line. Do a similar action with your back straight leg. Curl your toes under lift your upper inner thigh toward the ceiling and isometrically drag your leg towards your pelvis and in towards your mid line. Take a few deep breaths here then explore opening with your exhale and creating length in your legs and pulling in towards your mid line with your inhale. Breath into your lower back to create space there and from this new space open your upper chest woking towards balancing your shoulders.
According to Light on Yoga you do use the strength of your buttocks muscles in order to lift your legs. This is also recommended in a few other books as well. However with this said yoga has changed and been adapted for our Western culture that does not have the same lifestyle habits as those in India where yoga originated. If your buttocks contracts slightly as you lift your legs and there is no compression on your lower back then I would say great. If there is pain then what is probably happening is your large stronger buttocks muscles are taking over from the deeper back muscles such as the erector spinae and the quadratus lumborum. Ideally you have a balance during the posture where the deeper gluteus minimus and the larger maximus to a certain degree are engaged as well as the back muscles previously mentioned. May of us tend to solely use the buttocks muscles in the this posture and never develop the full potential and strength of the lower back.
Half Moon or Ardha Chandrasana is a beautiful and very challenging posture. If you are working on opening your pelvis to prepare here are some ideas that will help. Because the hips are opening away from each other in Half Moon, postures that do the same like Warrior Two, Triangle and Extended Long Side Angle will help deepen this opening for you. These standing postures will also strengthen your legs which is crucial in this balancing pose. You can also explore the opening of the hips in this posture by practising it at the wall or a table. Allow your front bottom hand to rest on the wall or other support and focus more on stacking the hips into one line.