Most research positions require credentials (like a PhD). The best way to get such requirements waived is to personally impress somebody. It would typically have to be somebody at the level of a research director of a lab, as in the case of your failure they are responsible for not following the rules.
As to self-study to a PhD level of knowledge, that's not too hard in math if you are disciplined and very capable. But in things where there is typically some experimental component it's more difficult. If you stick to areas requiring only computer simulation you can probably get away with it. However, it will take a long time. As a PhD you need to go deep for your thesis, become a serious expert in some area of endeavor. This takes years. So even if you're really, really good I think you have to allocate at least three years for each area. However, if you are just talking knowledge, as in enough to pass your quals, you can probably get away with a year or so for each area (if you're as fast as you say), maybe two if your undergraduate preparation is weak.
Sounds very ambitious. Good luck! And, by the way, you might want to pick up a law degree along the way to make it easier to patent things.
But I'm curious, how do you know you have "knowledge equivalent to someone with a BsC in EECS, physics, chem, bio and mathematics"? And do you mean equivalent to somebody good coming out of MIT or somthing else?
