I write as someone about your age (although married for many years). I'm a woman if that matters. My initial thought was that you had each had rational although different approaches. I would categorize yours as "generous" and hers as "practical". IMO either could be viewed as "fair." But then you said that a fallback would be for you to purchase a smaller/cheaper house and her position is that you should pay the mortgage without a contribution from her of any kind. Hmmm...that's neither practical nor fair. You may find that your future is that her money is "hers" and your money is "ours". Not a situation I would want to be in.
Catbert, brilliant as usual, has identified a key issue that you state without seeming fully to recognize: your intended partner in sin wishes you to pay for her living space. She does not wish to split expenses. She wants you to pay unequally.
I can only speculate what the significance of this is, but I recommend that you think it about specifically, and decide upon your response based on careful consideration. Here are my questions.
1. Is she viewing you as partly or primarily as a financial rescuer - a knight in dollar armor?
2. Does she view the woman's role as being desired lover, while men are (to be blunt) basically supposed to pay for the privilege of being with her?
3. Is she honest with herself about whatever her views are, or does discussing these matters require articulating things that she does not admit to herself?
If option 2 is the main thing, it's up to you whether you wish to enter the venture on her terms, but you need to be aware that's what you're doing. Living in her house and paying half the mortgage while getting no equity is a perfectly limited implementation of path 2. Buying or renting a separate place that she lives in for free would also fulfill 2, but in your shoes I wouldn't undertake the level of risk entailed in buying a new place. Taking the good offer to sell your own home might be a good idea... except that then you have ready cash that she might view as, um, "ours."
If option 1 is the main deal, and you think that a partnership means equal contributions at your age, tell her no.
If the unclear part of option 3 is the main deal, you have someone using relationships to cover her confusion. I'd stay away from that too. Just my opinion, though. Love is a many-splendored thing.
My family had someone in his 60s who got roped into 1, 2 and the unclear part of 3. He was so suckered by her that he proposed marriage. Lucky for all of us, she refused. In the end, a different "partner" appeared... one who had her own funds. Lucky again!
May fortune smile upon you...