Interesting link and certainly plausible. However, even if true, there would be no need to cover up his birth certificate and a phony draft registration is unexplainable no matter what the circumstances.
Well, I don't think it's unexplainable, exactly.
Prior to July, 1980 registration was not required. After Carter's EO re-instituting registration, practices were casual to say the least. Many, many young men in 1980 and afterwards, knowing that there was no draft and (probably) would never be one failed to register. Through 1986, there were only a handful of prosecutions against thousand and thousands who did not register. Later, it got tougher, with student loan consequences and tighter tracking, but in the beginning, the "requirement" was toothless and often either ignored or legitimately not understood.
So, suppose Barry did not register in July 1980, as he was required to do. Maybe he didn't because he was holding himself out as an Indonesian citizen. Maybe he didn't because he hated America and the military. Maybe he didn't know about it.
None of that matters. When things got serious in 2008, he OBVIOUSLY needed a forged registration, because the real one never existed.