The little I know is correct.
There’s no halachic requirement to give a get unless beis din orders one to be given. In the absence of such an order there is no basis to demand or attempt to force one to be give. And a beis din can only order one be given if the halachic criteria that a get is mandatory has been met. In the absence of that halachic threshold being met, even beis din cannot order one be given.
akuperma, if a court withheld a civil divorce, that would not help or hurt one spouse more than the other. So taking that approach accomplishes little as it has the same effect on both parties.