I was looking into a similar issue earlier today: Unable to add user to PM topic. In that case, the issue I’m finding is that regular users cannot invite users to a PM if the max invites per day setting is 0. For your case, it is the enable personal messages site setting that is probably the cause of the issue. When that setting is disabled, only admins can add users to an existing PM. Moderators cannot add users to a PM when enable personal messages is disabled.
Are you able to double check to see if admins are able to add users to PMs on your site? For this to work, the admin will have to be the user who initiates the personal message.
I am fairly certain that this behaviour is intentional, because it is written explicitly into the code. I’m not sure what the thinking is behind preventing moderators from adding users to a PM when the enable personal messages site setting is disabled. Possibly this is something that could be changed.
I’ll leave a note for our developers here, in case anyone wants to look at it. The issue is occurring here: https://github.com/discourse/discourse/blob/master/lib/guardian.rb#L357.