I would not suggest that you report a lawyer to the state bar unless you have good reason to do so and you haven't stated any facts which would suggest as such. I'm a trial and I can tell you many people suggest that as a first tactic. If you proceed in that manner without real evidence except your own perception of bias, that will not go well for you and could hurt your case.
My suggestion: do you have a lawyer? If you don't, hire one. If you do, consult with your lawyer as to how to proceed. If you don't like your lawyer, hire another lawyer.
I sounds as though you really need to hire an attorney to represent YOUR interests. With a Custody case, the parties are submitting the determination of custody to a stranger. That is why the concept of a GAL exists to help the Court make a finding of the best interest of the child. One party will inevitable disagree with the GAL. The parties could not resolve their differences and so now Judge must. I often hear Judge instruct the parties that if I make an order that you both hate a little then it probably a good order. So I do sympathize with you but you need to hire your own attorney and not attack the GAL.
A GAL is appointed by the Court to express their belief as to what the best interests of the children are. This may mean something entirely different from what you, your ex, or your child desire. The GAL can express that he believes that legal and / or physical custody should fall to one or both of you. If the GAL is not recommending in favor of your position, then this is why you need your own attorney and not complain about the GAL.
I would also suggest that you not focus on terminology such as "full" custody which actually does not exist. Custody is first divided between legal and physical, then we look at types of custody, namely, sole, joint, or shared.
With Legal Custody, regardless of what is labeled, sole , joint or shared, both parent have access to all records for the child and each parent must give the other notice of changes of address. So I would not wage war over any issue of "legal" custody.
With Physical Custody, now this is where the issues really lie. What is the physical schedule between both parents and who makes those day to day decisions in the child's life. This can be a very vague free flow of visitation back and forth with the child or a very rigid schedule. So here, You MUST focus on the child. Forget the legal terminology, look to what is important , the time you spend with your child. Once this schedule or relationship between both parents and the child is set , then everything else will fall into place. Kids, no mater the age, really do pick up on whether Mommy and Daddy get along or not. Be Careful not to fall into the trap of wanting to win at the cost of your child.
The essential answer to the question is yes, you can hire an attorney even if the child has a GAL.
Many people erroneously will refer to a GAL as the child's attorney. This is not the case. If anything, the GAL is the Court's attorney, but even that description is wrong. The Guardian ad Litem is responsible for investigating important issues related to child in order to provide that information to the court with a recommendation regarding what the GAL believes is in the best interests of the child.
If you were to hire an attorney, the attorney would represent the legal interests of the child. I do not think this is what you are really seeking. If you hire an attorney for the child, the attorney would consult with the child, determine what the child wants, and advocate for that position. What you would want would not even be factored into the equation because your child's attorney owes no duty to you, even if you were the source of the payment for his services. If your child agrees with the GAL position, for example, the attorney for the child would be required to advocate for the GAL position.
As indicated by the previous posts, it sounds more like what you want and need is to consider getting an attorney for you, not your child. If it is YOUR position that is not being properly presented by the GAL, then you need an attorney dedicated to presenting YOUR position to the court. Hiring an attorney for the child will not accomplish that.