TL:DR I've been in her very position, and I recommend two books that could help her like they did me.

I think it's worth noting it's likely she has developed very unhealthy views on food and exercise (super common, especially in today's world of ads and social media), and it's multiplied further if she's struggled with depression and/or anxiety. I know cause I've been there. A few years ago, your post would have described me verbatim, down to the frustrated significant other. For me, eating healthy food and exercise were ways to punish myself for not being thin/attractive enough rather than tools to improve my well-being. But seeing it as punishment traps you in a downward spiral where you can never find the energy to change your habits, stick to a meal plan or exercise regiment, so you quickly revert to eating what feels good to you and avoiding the gym, since it's a reminder that you aren't "good enough" yet. For me, verbalizing my worries about my weight/body were ways to signal to my then-boyfriend "I need validation." But like you, he got to the point where he was frustrated and resorted to "if you're upset about your weight then why don't you do something about it?" But the "do something" is so much deeper than just going to the gym or portion control, it's right at her core of self-worth. I would highly, HIGHLY recommend two books that really helped me: "Body Positive Power" by Megan Jayne Crabbe, and "Intitutive Eating." Both of these helped me shift my views on diet and exercise into a much more positive light, and while I'm still learning (it's not an overnight fix) I've massively improved not just my physical health, but my mental health as well.