The reality is that most women don't support that. It is the typically loud feminist minority that rammed this through sympathetic leftist ears in the Administration. The "they" you want to punish by requiring them to register for the draft are 18 year old girls who had nothing to do with that asinine decision.
Heck, most of the Women Marines I knew, even officers, knew that infantry should remain all-male. It is a few loudmouths who try to change that for their own ego/career aspirations.
What Republicans need to do is to America's 18 year old girls and their families what the left really has in store for them. If you want a real "War on Women", making them subject to the draft an involuntary placement in combat arms units is a pretty good start.
Then the women who didn’t support the push by the feminists should have stood up and argues against the feminists. But they didn’t. I would also disagree with your statement that it is “punishment” to require someone to register to possibly serve their country. Nothing says they will be called to serve just like it is for men. And, if called to serve, they don’t have to serve in the combat jobs. So, appears we have a fundamental difference of opinion regarding whether women should be required to serve their country — just like men. To me, equality means equality in all ways, not just the “good” things that they want.