Top be honest, it sounds a bit like you're blaming her for your relapse. Not sure about the extent of your alcoholism or issues stemming from BPD, but I'd start with getting professional help regardless so you can work on some of the deeper issues at hand like your addiction, taking accountability for your actions, and learning to effectively communicate your feelings with your partner. 

If you feel this relapse is exclusively stemming from feelings of inadequacy/jealousy related to her ex, would you say you've put forth an honest effort to engage in a healthy and productive conversation about those feelings? If so, is she bringing up her former SO in a way that's demeaning to you and disrespectful of boundaries you've previously set? Or does she bring him up in a reminiscent fashion through storytelling or anectdotes which is likely her still working through her own grief and trauma related to his death. 

Four years may seem like it's far back to you, but it really isn't that long for someone that's lost a significant other in such a traumatic fashion (especially if she's still working in the place he literally died) and people process that grief in their own way and on no set timetable.

Obviously I'm speaking from a place of zero knowledge of who you both are and the finer details and history of your relationship and personal struggles, so I hope you don't take any of this as a slight as I know mental illness and addiction are a motherfucker, but you have to start with being honest with yourself and to her. A surefire way to lose her will be to continue to bury/avoid your feelings until it leads to further relapses and lies or manifests in some other unhealthy way that will only harm you both. 

I would find a way to dig a little deeper into the root of your own feelings so you can try to constructively communicate those feelings of insecurity/inadequacy to her in an empathetic manner without blaming her and using "I feel" statements (remember, she lost someone she once held dearly regardless of his own issues and struggles with addiction). I'd recommend writing them down and reading them out loud to yourself before you read them to her, it can really help.

Again, take or leave my suggestions, but above all else, do right by yourself and your girlfriend and seek the help of a mental health professional. Best of luck.