snap packages themselves live in /var/lib/snapd and store their non-user-specific data in /var/snap …
snap uses apparmor for the security confinement, apparmor does not deal with cross-disk symlinks … but you can use bind mounts … so copy the contents of /var/snap and /var/lib/snapd to the big disk and create bind mounts to the real /var/snap and /var/lib/snapd dirs … (the same also works with ~/snap but you need to make sure to get the user permissios right at the endpoint of the bind mount)