Like most things, the principle is fine...sort of. It's when human's start to use it, is when it all falls down.
Like Cogito says, how do I prove I'm in a relationship with someone? When am I allowed to run a check? Let's say I am a single mum and I meet a guy at a bar. I snog him. Am I in a relationship? Should I be allowed to run a check now before I introduce him to my kids? If not why not? Do I have to sleep with him first? Or somehow prove I've been seeing him for a month...perhaps 6 months? A year? Two? Isn't the latter too late?
What evidence do I have to produce to prove I am having a relationship with said person? If none, whats the stop me from running random checks on a guy I happen to not like but is going out with my say.....sister/BF/auntie/cousin/whoever? Surely we should be entitled to some privacy?
Just what are the police allowed to tell the requestor without infringing on his rights to privacy? Let's say I beat my previous GF up. Are police allowed to say "He beat his exGF up." or can they only say "He has a conviction for assault".
Regardless, if I was an abuser I'd lie my arse off anyway. I'd claim it was all a big misunderstanding. She was crazy, the police made a mistake. She lied to get me back and I was done up like a kipper.
Abusers are highly manipulative. I can see it already with someone I know (fortunately not that well). Her new 'partner' is slowly isolating her from friends & family and making him the centre of her world. It doesn't matter what the few people she sees says. She won't believe them anyway and will simply take what he says as truth. Eventually she'll believe she's lucky to have him in her life as noone else cares about her. Police background check? If him having 9 kids with 6 different mothers and a 20+ year age gap isn't enough of a red flag, i doubt a little report from the police will have any impact at all.
In short, fine in theory. In practice I fear it's full of holes and open to abuse.