A regular family counselor is more than likely going to be either a psychologist, or a LCSW (Licensed Clinical Social Worker) anyways.
I don't know if you'll go in with him or not. I went in with my son only because he asked me to. I would have let him go in on his own. Of course, I talked to the psychiatrist alone after he was done talking with my son, and I imagine that's going to be pretty much what you will do too. Well, probably before... they have to know the reason they are seeing the child.
Whether it would weigh more in the eyes of the judge no one could really tell you. However, in my layman's way of thinking, I'd have to say that documentation by a professional would carry more weight than that of your written notes saying he told you this or that. Then, of course, you'll have to get into the whole Rules of Evidence issue... whether it's admissable or not, heresay or not, etc. if you all decide to bring any of this up in court.
If the child is having issues with all this, then by all means get him in to see and talk with someone about it. But don't let the fact that "it might burn the ex in court" or "prove our case" be the motivating factors behind your choice. Not saying that's what you are doing mamadi, just throwing that out there for all to see.