Trouble is, there is no such thing as a "management consultant degree". You know this, and so does DHS. Again, you are saying nothing about the actual job you are going to do. You could be an MC in product developement as easily as bus scheduling. A BA. or even an MBA wouldn't be a satisfactory prerequisite, say, marketing, or Urban planning. You can get an MBA with a Bach in any discipline, so the job becomes the crucial element. Remember, for MC in the TN world, you are being hired to solve a particular issue, you are not being hired on as a permanent Management consultant, as the position is supposed to be supernumerary to the firm's normal org-chart.
Thus the need for a job which closely matches your education or your experience. Don't know why you are being evasive about the job. I would trust you won't be so tight-lipped at the border (or in the firm's TN letter) or your application is doomed.
Now, once you have worked as an actual MC, then your 5 years of MC work can qualify you for just about any type of MC work, but without those 5 years as an MC, you are kinda stuck doing what your education makes you an expert in (that is why I said earlier that your transcript, with its course load, may come into play).
So, are you missing something? Don't know, but you certainly aren't telling everything.