Can my ex be prosecuted for signing a state driver's permit for our minor as sole custodian when the form required both of us to sign because we have joint custody? He literrally lied and signed as 'sole custodian.'

Best is to call and ask.

Most likely no, you may try and take them back to family court to see if they will hold her in contempt for not allowing you to be part of it.

*** get over it, try and work together, why not be glad the child got the drivers licesne, and why do you want to ruin the event of your child to get back at the mother.