OK, I've found a way to get this to work, but without any understanding of what I'm doing.
I went into bridging mode, then in the virtual XP machine used the "Add a network place wizard", setting it to aim separately at the EXT4 and NTFS drives on the Linux machine by entering their network directory names. This brings up the drives as visible destinations in XP's "My network places".
For some reason, after doing this, I can now ping the Linux machine from the command prompt. This definitely didn't work earlier in bridging mode.
I can now set up the automatic backups, so the stress level has come back out of the danger zone. It woulld be nice to have an explanation of any of this, but I'm probably too old!