Do graduate school admissions committees care about an official bachelors degree, or just the relevant coursework?
Short answer: You will almost certainly need an official bachelor's degree.

Long answer: Your main obstacle here is not the admissions committee, but the requirements of the university you'd be applying to. Any university in the US surely has it written down officially in multiple places that Masters and PhD students must hold a bachelor's degree (or its foreign equivalent) before enrolling. I believe doing otherwise may even get the university in trouble with their accreditation board. 

Even if there are exceptional circumstances under which this rule can be broken, the university would have to really want you. "Pretty competitive" won't cut it. And indeed, why (from their point of view) should they admit you now if they could admit you in a year, after you have more training under your belt?