What you are saying is essentially at its core, true. MH diagnoses are indeed, simply labels. They do not mean that you are capable or incapable as a parent. There is no proof, anywhere, that having a mental health disorder causes you to be an inadequate parent.
The only exception to this may be anti-social personality disorder where the nature of the disorder often shows patterns of harm to children and family relationships, but that is not the diagnostic issue here.
Many good parents have mental health histories and diagnoses and this should not, unless neglect or abuse occurred with the children, affect the decision of the case.
It would not only be short sighted on the part of the court, but it would be discriminatory as well...As a matter of fact, you have a strong case to say that the MH diagnoses that you have been given are being used as a way to discriminate against you as far as custody of the children.
It really isn't about you proving your diagnosis is the issue; it is the court's issue. They have to show that behaviors that occurred on your part are the reason you are capable or not as a parent, not a label.
How do you combat the label? You should be very assertive about the fact that there is no proof that having an MH diagnosis is a reason to prevent joint custody. I would ask instead for a list or specific behaviors that you have done as evidence instead. Anything less is discriminatory and is against the law. Period.
I would also think about contacting NAMI, the national alliance for mental health and seeing if they can help provide information about MH discrimination. Why fight an accusation when you can fight back yourself? Use the diagnosis label against those who are using it against you! What they are doing is wrong. A person is not a label and that cannot be used as a basis for custody, only actions.
On a personal counseling side. You do need support throughout this difficult process. You are stressed and frustrated and need an outlet.
Many forms of counseling such as Gestalt therapy do not label clients, but rather simply help them in the here and now. You may wish to see if there is a Gestalt therapist in your area, as this would be a much more pleasant (and non-diagnostic) experience for you.