According to https://wiki.filezilla-project.org/Howto#Other_platforms, FileZilla works fine with standard SSH agent. I tested it on Ubuntu with XFCE+GNOME-keyring and verified that I didn't need to do any key file format conversion. The PuTTY format is only necessary on Windows because FileZilla uses PuTTY library on Windows; but you mentioned you were looking for a program for Linux.
The usual combination of tarballing, compressing, and scp. It's worked perfectly for several decades (and no issues with port behavior and nasty NAT fixups), so- works for me!