Two tiny suggestions
Hey, I did some more reading on CMT and I may have been wrong about being like your mother as some places said that and others said that each person is different even in the same family. But CMT is quite common and it is rare for people to lose ability to walk with this.
First suggestion–google Charcot Marie Tooth and there is a Charcot-Marie-Tooth Association kind of like this one that has some good information.
Second, if there is a chance to argue about the insurance paying for the testing, the best argument is that if this were CMT, you would not need immunoglobulin infusions any more. Since these are really expensive, the insurance might would pay for the CMT testing in order to no longer have to pay for the immunoglobulin. I am not being cynical–just practical in that that is the way the world works–if you point out why it is a potential huge benefit for THEM, they might do it.