Marketing PhD would really fit your profile, in my opinion. More specifically, Quantitative Marketing (which, basically, is the use of economics and statistics in Marketing).
It seems to me that the PhD program you mentioned at CMU is not at Tepper School of Business, but Dietrich College of Humanities & Social Sciences. If you apply to programs outside business schools, they will probably not accept your GMAT and you should take the GRE instead. Please confirm it first.
Remember that PhD is different from MBA. Extracurriculars may not be that relevant, unless they help in research, for example.
If you really get a GMAT score of 750, it's a great score, specially if you ace the Quant section. It should be enough to make any school take a look at you, even the top ones.
It seems the lack of research experience is a weakness in your profile. I see a MSc in Chemistry in your profile. Didn't you do any research over there? Even if it's in a different field, it would help.
Some people try to get some research experience working as an assistant to a professor, for example. But i don't know hat that works.
I don't see a problem in the disconnection between education and work. I don't think they are even so disconnected. There are lots of engineers doing Marketing PhD and consultancy is a possible work for them. If you apply to a PhD program whose reseach in done in partnership with companies, to solve their actual problems, experience in consultancy may be interesting. Just don't state that's your goal, since your aim when applying to a PhD in a career doing research in academia.
I think many posts in my blog can help you, like this one:
https://myphdtales.blogspot.com.br/2017/ ... y-for.html