Except that the principle will already have been established in law that women are liable for the draft. To then exclude them would be to make an exception of the then established law. And what is the point in having women register for the draft if they would be exempted from any real draft?
It would do no such thing. It would establish that women are eligible for the draft, it would not force women to be drafted. The whole point of having a selective service registration is to have a ready list of names that can quickly be used in time of crisis. If someone sues to include women in a draft that Congress hasn't authorized, a lawsuit will take too much time to be any good. By the time it winds through the courts, the crisis would have been over.
And what is the point in having women register for the draft if they would be exempted from any real draft?
The point is to have another ready list of names that can be used in time of crisis. What is the fastest way to get MEN to the front in combat roles? Draft 18 year old men and put them through training, or draft women to backfill stateside clerical roles and move men who are already trained to combat units?
Advocating women to register for Selective Service is not advocating dragging women away from their newborns and putting them on the front lines.