Yes, they will try to persuade you not to change. But if your employer changes NHI contributions to new ID your card will be cancelled.
You can just insist on the change. In your case you probably can wait until you get the Taiwan ID and ignore all of this new ID nonsense.
You can show them the form 被保險人變更事項申請書(勞保、健保及勞退3合1表格)