Ultimately the bank will have first call on the house and you will be the only one on the hook directly to the bank if you don't make the mortgage payments. There's nothing you can do to avoid that if you can't get a joint mortgage.
What you could do is make a side agreement that your girlfriend would be entitled to half the equity in the house, and would be required to make half the payments (via you). You could perhaps also add that she would be part responsible for helping you clear any arrears.
But in the end it'd just be a deal between you and her. She wouldn't have any direct rights over the house and she wouldn't be at risk of the bank pursuing her if you don't pay the mortgage.
You'd probably also need legal advice to make it watertight, but you could also not worry about that too much and just write it all down as formally as possible. It really depends if you're just trying to improve your feelings about the process or whether you really want something that you could both rely on in the event of a later split.
I don't think getting married would make any make any real difference day-to-day. In law, with rare exceptions, the finances of spouses are independent from each other. However in the longer term, being married would mean your now-wife would have a stronger legal claim on half the equity in the house in the event of you splitting up.