Hi Whozzat
Sounds like using any filesharing program will be very difficult with your set-up, especially since you wouldn't want to tie up the phone lines.
Call/ask around and see what Internet Service Providers (ISP's) are in your area. You can get "always on" service at pretty reasonable monthy rates.
See if
http://www.broadbandreports.com/town can help find you a decent ISP that alllows file sharing.
Cheers