I doubt it. If you look at the definition of a J1 visa it doesn't talk about any prior training precluding you from taking more. There may be some limit on the J1 program run by a particular school I suppose, but in my case, the US Immigration guys didn't ask about my experience and qualifications at all....
However, the schools that provide the J1 do so on the basis that they have a captive customer ready to drop a lot of money on them. In your situation, the school may feel that an ab initio CPL(A) is worth more than a CPL(H) conversion, and therefore not wish to sponsor a J1 for you - they'd probably be happy with an M1 though.
Best thing is to talk with school(s) in question.
BG.