A register is exactly the kind of thing that is needed but would be extremely unpopular if suggested I think, so probably a no go, plus, would it actually deter women from getting into a relationship with somebody? Especially if the introduction of it had been controversial and had a lot of backlash. But it is a really interesting thought experiment - why isn't there a register like there is for sex crimes?
There are clear patterns of male violence and I think it would be worth more follow up from incidents in the lower parts of the pyramid.
The court system needs a whole overhaul - women cannot protect their children from violent fathers and that exposes children to more trauma. Not all traumatised children will go on to become abusers, but some will. And all children would benefit from less trauma.
Social services needs proper funding and resourcing. Again not all traumatised children go on to be abusers but all of them would benefit from better support and intervention.
The benefits system should actually be a safety net so that women can leave violent partners. Refuges should be plentiful and safe havens.
Free and abundant mental health support for all on the NHS. Might help some would be abusers to change. Would help victims to leave and make sense of what happened to them.
All police, social workers, family court workers, healthcare staff, teachers given full training in endemic sexism, male violence, toxic masculinity, domestic abuse.
Currently most people, including the ones that really really need to, don't understand the mechanics of domestic abuse. And very few people even understand the existence of endemic sexism, sexed male violence etc. I think that's why most people think it's bigoted to object to transwomen having access to female only spaces.