Fortunately, the customer bought Ikea flat-packs which are always good quality. Sometimes people find similar products on-line and they don't measure up. If you are considering a project like this, spend the extra for a quality product. Whatever extra you spend on the product, you'll save in installation.
With a stud behind each box I could screw directly into them. And I added a hollow-wall anchor to level the bracket. Not everyone is as concerned with symmetry as this client, but he was pleasantly surprised that it came out exactly as he'd imagined.