Is it possible if I got a "converted" PPL then take the training there and get the full FAA CPL?
I don't know specifically for the FAA, but here in JAA-land, the entry for CPL is any ICAO PPL. Doesn't have to be a JAA PPL.
Might be worth checking that out. That would mean that you do not need to get any form of FAA PPL (standalone or piggyback), but do the CPL flight test straight away.
That is, if you don't plan on exercising any PPL privileges in the US in the runup to your CPL exam.