UPDATE: Now working.
After a lot of messing around, the following steps managed to work:
1. Remove the additional command I added via ISPConfig "Server Config", in the "Jailkit chroot app sections" and "Jailkit chrooted applications" boxes.
2. As root on the server, edit /etc/jailkit/jk_init.ini
Under [basicshell], appended the following to "executables":
3. Restarted jailkit
4. Added a new website via ISPConfig
5. Added a new Shell user, zip and unzip now working