I'm reminded of a time growing up when my mom would tell us to behave and I found myself wondering how I could possibly know when I was "being have."
Joking aside, there is something you mentioned that I find might solve your dilema about what to call mental/behavioral health. The fact that external factors sometimes affect this health, or that underlying, changeable or unchangeable factors certainly contribute, but, all things considered, the question essentially has more to do with simply giving people guidance and assistance not with how to be mentally healthy or even with how to behave but with how to be, or simply being. We're not given a user's manual with our existence; if we're lucky, which few are, we'll figure out how to be without causing too much damage in the process, however, for most of us, we never quite figure out how to be and life ends up being a series of trials and errors often with harmful repercussions. Behavioral health conjures images of dog training or potty training, a rather pavlovian approach to this confounding notion of how we are to exist.
Really, behavioral health simply describes acting out one's existence in a manner consistent with the expectations of society, it means conforming, it means following the rules whether they are apt or irrelevant, appropriate or dangerous, correct or ill-advised and honestly, conformity can have negative repercussions on one's mental health (especially considering that homosexuality is considered bad behavior by a large portion of the population, yet feels the true course for some). And indeed, mental health is not fitting for habits or addictions that are detrimental to one's health. Really, I think what you seek is a term that might describe the guidance, knowledge, practices, and tools required for one to derive the greatest benefits from life while minimizing the harm and dangers (to self and others) that might result from living recklessly, carelessly or unhealthily. Perhaps Mortal Health or Life Health...yeah, I understand your connundrum in naming this notion, because really, you seek to place human existence into a nice neat little package that will apply to each and every; however not all creatures that claim human sentience can fit into one particular category, what is defined as "broken" for some is quite normal and beneficial to others (I refer here to my own often pessimistic outlooks, my tumultuous childhood, my borderline insanity and breakdowns and the poetic and artistic insights that suicidal thoughts have given me and have given artists throughout history.) Also, some measure of abnormality is required in order for us to come to a decision on what defines normality, therefore, if psychologists were to succeed in curing every ailing mind or behavior, they would effectively recalibrate our notion of normalcy, because without the broken, you cannot know what is whole.