Actually, there is a TCP/IP solution for Workbench 1.3, but it's very limited in its flexibility. It's called AmigaNOS, an Amiga port of the Network Operating System program, originally released on the PC. It's a standalone environment, which incorporates SLIP/PPP, the TCP/IP stack and several applications that you can use from within the AmigaNOS environment like FTP, which is what you're interested in. It doesn't have an API, at least none that I know of, so there's no way to use software that expects a bsdsocket interface (even if you would be able to run it at all under Kickstart 1.3). You're stuck with the features/programs the program has to offer.
I've unsuccesfully tried to get the program running in the past, just for the heck of it, but couldn't get the PPP connection working. To make things more complicated, there are several versions of AmigaNOS floating around, some of which support SLIP but not PPP. There's a version of AmigaNOS on ye ol' Aminet, IIRC. Good luck if you do give it a try and let us know if you have any luck getting it to work!