Prestige for MBA matters for some opportunities - but if one has a great internship (like my DD did) - it is a global corporation. So she has her foot in the door already.
IMHO if you JPG do go to UA, stay in STEM MBA as long as you can. You may get a great co-op or intern opportunity that will get hour foot in the door to some fabulous opportunities if you shine in Eng and in business. Plus you will complete it much earlier! One extra year VS two…
Also prepare oneself to be well-rounded social and many other areas. Climbing the ladder in any corporation - keeping the eye on quality work, group efforts, etc rather than solo star - maybe call it using the golden rule in the workplace.