The easiest solution is to place the shared mailbox on hold. Microsoft 365 retention policies can be used to achieve the same, and allow more customizability, but are a bit harder to configure.
As for disabling the "purge" option, there is no way to do that for a user that has been granted Full access permissions. You will be able to restrict it if you leverage folder-level permissions instead, but that's a hassle to configure.
Also have in mind that both litigation hold and retention policies require Exchange Online Plan 2 license on the mailbox.