Having delete anywhere that people look at it all of the time is probably bad. That's a super special case button that should almost always route to a manager for approval and should be trivially disabled as it often would need to be forbidden.
Along this lines, perhaps this should be a setting under Group Permissions.
This could definitely be a good route to go with this.
I would think not changing the organization of the ticket would be a good thing in most cases.
Add an option, but also a confirmation.
Are you sure you want to change this tickets Org to " "?
"Most cases." Problem is, if there are any cases, it needs to be able to be changed. You don't want ticket data going to the wrong entity. I can't think of any case where it would be okay to be locked.
That was my point, allow the change, but confirm it before it's set. You don't want a ticket getting moved from one org to another org with sensitive information in it, by mistake.