Yes, it's better to install under user's home directory - they do need RW permission for zimbra directory.
By the way, in your case, you may also want to disable auto update for your users - since updater also prompts for password. Look for this file: extensions/prism-runtime@developer.mozilla.org/defaults/preferences/preferences.
js, and set this key to be false: pref("app.update.enabled", false)
Allowing non-admin user to run ZD is a feature that's already on our road map.