Get 10 new $100 bills. Cut them into thirds. Put all of the left sides in one envelope, right sides in another, and middles in a third. Send the envelopes separately.
1/3rd of a bill is not legal tender. Therefore, each shipment has no value. However, banks can reissue the money as long as you have > 51% of the material for each bill. Therefore, if one of the shipments is lost, the recipient still gets $1,000. 
You can fully eliminate the risk of loss (and reduce shipping costs) at the expense of latency. Wait for the recipient to acknowledge the receipt of each shipment before sending the next one. If a shipment is lost, split the bills into smaller parts so that you always ensure that you and the recipient have more than 51% of the bills. Once the recipient has 51% of the bills, you can stop making shipments, saving you costs.
Its great seeing answers from someone else. I love a different perspective. My default answer is don’t send the money through the mail. Instead you should make a physical exchange, but sometimes a person can’t do that. What would you have come up with?