If you let something as small as a missing bag containing easily-replaced items "ruin the vacation of a lifetime" you are never going to enjoy much of anything.

1. When a bag goes missing, simply complete the required report and leave the airport. If there are additional details, call afterwards, but all that is needed on the spot are the basics which you have. LATAM also permits you to make a report via its Contact Center within 7 days of arrival (international). You will be assigned a tracking number which you can use online.

2. Stop worrying about the bag. Nothing you do is going to make it appear any faster. Do not change your travel plans, miss tours and the like. Under the Montreal Convention, you are entitled to reimbursement for reasonably necessary expenses up to a maximum of approximately US$ 1,650. This sum will be deducted from the value of the bag and contents should it ultimately be lost or damaged.

3. Call the Contact Center. Unless there is a good explanation for the delay, advise that you will file a DOT complaint (I presume that the ticket was purchased in the US) if you do not have full payment within 7 business days and that you will initiate a chargeback for the funds against your CC.

As an aside, a much easier route is your travel insurance. You would have a check in hand and your insurance carrier would collect back from LAN and you would be done with it.