Thanks BigStar for taking the time to help me.
Both my source and destination FTPs
are using SSL/TLS, so it should make no difference to NAT, if I understand you correctly. Yet other people are using FlashFXP to copy between sites, the only thing different I can see is that I use NAT and they don't.
Maybe I should try just bypassing my NAT router here and see if everything works with a direct connection. Then I'll know if it's the router or not.