As the original poster is in New Zealand I can not comment on what licencing system they have in place, but normally you would have to do a differences course that would cover the engine and it's interface with the airframe, so at a guess what you have on the 737 has no relation to the 320.
I would not let that however stop you from applying for the positio, not everybody has everything, and companies do offer training for the right candidates.
It seems some are basing their arguements based on knowledge, or lack of, on the EASA system, for those of us that hold various licences and authorisations from differing authorities, and company autorisations under a company approval as issued by an NAA different to national one. It is possible under some systems to hold limited authorisations, I know I certainly have in the past.
