Here are a few ways to discredit the custody evaluator at trial:

1. Bias. In personal injury cases, the insurance company will hire a doctor to examine the injured. Insurers pay thousands of dollars (now you know where your premiums go) to certain doctors who are prone to rendering an opinion favorable to the insurance company. These "independent" experts are often discredited by the plaintiff's lawyer bringing out the hundreds of prior opinions these physicians have rendered against injury victims. The same holds true in family court. Most custody evaluators have years of experience and have rendered hundreds of opinions. If there is consistency in those opinions, they carry a bias. Certain experts are prone to rendering certain opinions. Make the court aware of the bias of the evaluator and the recommendations may be discredited.
2. Diligence. We've cross-examined custody evaluators who have spent less than an hour in the presence of our client and the children that are the subject of the action. How much can anyone learn about a familial situation in 60 minutes of observation. What if the kids were having a tough day? What if the parent is nervous about the scrutiny of the evaluator? What if the dog won't stop barking? Think of it as a movie. If someone stopped "Titanic" before the ice berg and never watched the ending, they'd think everyone arrived safely in New York and wouldn't know the whole story. Evaluators are busy people. That haste can be taken advantage of.
3. Qualifications. Just who is the evaluator in your case? Do they have Ph.D.? How many evaluations have they conducted? Who are they employed by? What is their degree in? Have they been subject to an action for malpractice or ethics complaints? Disciplined by a professional board? Are they a licensed psychologist? All of these questions go to the foundation of the expert's opinions. Get them disqualified as an expert and the court cannot rely on their recommendations.

These same techniques can be used to discredit other court-appointed custody experts, such as a Guardian Ad Litem. No kidding - we had case in which the adverse Guardian had a degree in art history and failed to spend a single moment with our client in the presence of our client with the children (despite a statutory requirement that she meet with the parent in the presence of the children in the relevant home). We attacked her opinions on all three of the grounds referenced above.