You sound foreign. A British man would have his own money and decency to know if she paid the huge lump sums into the house, she owns it.
Why would a British man have his own money? Many don't. And she chose to give him a half share in the house. Any man, British or otherwise, can reasonably conclude that they own half the house, which is what the law says and, in all honesty, is the moral position too. If someone gives you a gift, it is yours.
I have no idea why you think OP is foreign.
She will fight this tooth and nail if lawyers get involved and you will be left with nothing after bloodsucking lawyers are involved. In fact, once the lawyers take their share, you could be left with a large bill from the lawyers.
If she chooses to fight this, she will lose. She will then have to pay OP's legal costs as well as her own. That may not cover all of OP's legal costs, but it will cover most of them. Anything he has to pay will only be a fraction of the value of his share of the house.
I would do as she says and move out. You’re not paying her rent at the moment and the relationship has broken down.
Why on earth would he pay her rent? It is as much his house as hers. She gave it to him, remember.