This patient sounds more help-seeking help-rejecting than masochistic. the distinction may be subtle since masochistic patients may reject the helpful suggestions of others, and may not respond well to the interventions of the therapist or psychopharmacologist, but I think the underlying motivations are different. my psychoanalytic supervisor in residency used to say we are all masochistic (i.e. that everyone at some point has some need to surrender, even sadists) and in my experience, masochistic patients (who believe they are unworthy, deserve to suffer, hold up their suffering as intrinsically good) are a joy to treat and can do very well in psychotherapy. there are obviously different types of masochism, but I find masochists do their homework! help-rejecting complainers not so much.
help-rejecting complainers may have masochistic elements to them, but it is more likely they are borderline, paranoid, or covert narcissists. Typically, they need validation for being "the most difficult patient" and a "hopeless case." It is only when you gratify their need for recognition as experiencing the most intractable suffering you have ever seen and when you question your ability to even help them, that you can make some traction with treating them. In contrast, I would conceptualize the underlying motivation of the masochist to be to demonstrate their moral superiority (what Freud called moral masochism), or to maintain attachments (if the patient believes the only reason others show interest in them is because they are suffering).
It is also important to remember that even with all the other bogus personality disorder diagnoses in the DSM the concept of self-defeating personality disorder never made the cut, and for good reasons. psychoanalytic notions of masochism can be useful but feminists in the 70s and 80s were quick to point out that there was the potential to blame victims for their abuse. Thus even if you find it helpful to formulate patients in this way, I would not put anything in their medical record re: masochism.