Even though you are 'convinced', the problem as I see it is no one else is - at least no one that matters. In other words, you think mom is mentally unfit but she has yet to run butt-naked through the local Wal-Mart with a butcher's knife screaming 'we're off to see the wizard' thus effectively removing all doubt as to her mental state. Also, it sounds to me like 'we' isn't either parent of the child - as such 'we' may not have standing to stick your nose in someone else's business - requesting evaluations and mandates and such. However, you may under certain circumstances petition the court to have someone involuntarily committed for an evaluation. You could also do what a lot of people in your situation do, which is to use DSS/Child Protective Services as your own personal bulldog to sic on mom to sniff around and see what turns up (assuming she is stupid enough to let them in without a Court order - CPS personnel are like vampires - they have to have your permission to come into your house absent a Court order). Further, if she is committed, the only significant difference (besides the cost) between an initial psychiatric evaluation and an initial psychological evaluation is the pedigree (or lack thereof) of the person doing the evaluation. Your assumption that a psychiatric evaluation is superior or preferable to a psychological evaluation is kinda like putting the cart before the horse. A psychological evaluation is generally less expensive, makes use of time proven standardized tests - which are incredibly good at providing a diagnosis - which can then be used to determine if your a full blown glorified psychiatrist is necessary. If it turns out mom is not fit, father (not 'we') stands a good chance of petitioning the Court to have custody awarded to him (assuming as you suggest, that he is not mentally ill and would provide a better environment for the child).
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