To be honest, it sounds like you have caregiver burnout. It also sounds like if there's improvement, the relationship may be salvageable. Is there any way you can convince Becky to stay with her parents or a friend for a few days and go on a mini-vacation with your kids? Caregiver burnout can color the way you think and make decisions, so stepping out of the caregiver role for a bit can help you feel less stressed about the situation, and therefore help you approach the situation from a more rational and thoughtful viewpoint, rather than a reactive viewpoint. If you end up doing this, I'd recommend some sort of activity that takes you off the grid, like camping. I also recommend getting therapy for yourself, regardless of whether you divorce Becky or stay with her. A therapist can help you work through your emotions and develop appropriate strategies for dealing with social problems based off your needs.
With that said, you have two real choices, and I'll give my advice for both:
Divorcing Becky is the easiest way for you personally to resolve this conflict. It's also the nuclear option; once you start the divorce process, there's no going back. Once you've decided to divorce, you aren't obligated to do anything for Becky, just what you want to and feel comfortable doing. If you decide to divorce Becky, I would recommend that you give her closest family member or best friend advance warning, or tell them immediately after telling Becky, so that someone in Becky's emotional support network is ready to support her. At this point, you're no longer responsible for Becky's wellbeing. I would also recommend sending your kids to their mom's place or to a family member's place when you tell Becky. Breakups/divorces tend to be really stressful for kids, so this approach minimizes that distress to your kids and allows you to discuss the divorce in a (relatively) stress-free, age-appropriate manner with each kid. Your kids may want to remain in contact with Becky, so have a plan for that.
The other choice you have is to attempt to work things out with Becky. This approach would be a lot harder, but divorce is always an option if you don't see improvement. First, some things to think about: + How well do you understand EDs? Have you done any research into Becky's EDs? If she doesn't feel that you understand her EDs, it'll be hard for her to open up to you about them. + How do you react when she relapses? If she feels that telling you will provoke a negative reaction rather than a constructive reaction, she may not want to tell you to avoid that negative reaction. Your burnout, combined with Becky's avoidance, can create a self-perpetuating cycle. + Related to the first bullet, people with EDs often have cognitive distortions that affect their perceptions and the way they think. In other words, people with EDs view the world, particularly around food and themselves, differently than people who don't have EDs. Long term treatment for EDs is essentially training yourself to ignore the self-destructive compulsions caused by anorexia's cognitive distortions. It may be worthwhile to look at some pro-anorexia sites to better understand the cognitive distortions caused by anorexia, so you can recognize them when Becky expresses them. Pro-ana sites are extreme, but recognizing the extreme cognitive distortions in pro-ana sites will help you recognize their more subtle forms in Becky. + One of the most common reasons why people develop anorexia is that it is something they can control in their life. EDs are often rooted in anxiety, and can be a maladaptive coping mechanism for anxiety/stress. Is Becky often anxious or stressed around her relapses? If so, anxiety/stress could be a trigger for her relapses. What else could trigger Becky's EDs? Next, you need to decide what your absolute requirements for a functional relationship with Becky are, moving forward. + From your post, kids coming first is one of those requirements. + Continuing treatment for her EDs should be a requirement. If Becky's current therapist isn't working for her, request a different therapist from the treatment center, or find another ED specialist outside the treatment center. The key to effective therapy is the interpersonal relationship between the client and the therapist. If the interpersonal relationship between Becky and the the therapist isn't working for Becky, she's wasting both her time and the therapist's time. + Anything else you think would be good. You know your situation better than we do, but fewer is usually better with these sorts of things. Once you've come up with your requirements, you need to have a conversation with Becky. Here is a rough script: "Hey Becky, I love you a lot, but lately I've been really struggling to help you with your ED. I feel that I've had to give up on a lot of opportunities for my kids to help you, and it feels like my attempts to help you haven't been successful. I can't remain in our relationship the way it is now, so what can we do as a team to help you with your ED?" Here are the important points of this script: + Reaffirming your love/positive feelings for Becky + Stating the problem (the interference Becky's ED is having on your life) + The reason(s) why it's a problem + What will happen if it remains a problem + Setting up a collaborative approach to solving the problem This script can also be adapted to starting the conversation for divorce, just change the last sentence to "I can't remain in our relationship, so I need to divorce." I would also recommend couple's therapy after this conversation. If she can't confide in you when she is struggling with her mental health, you have a communication issue. Couple's therapy can help with that.
Best of luck!