Insurance for expensive items in carry-on moving from US -> Canada

I assume that if you can take these antiques in your carry-on when you fly to Canada then they are reasonably small. If so, you can buy USPS insurance for up to $5,000 to protect against loss or damage. You can view the cost of the insurance here.

There are also provisions for Special Handling for fragile shipments including unusual items like bees, animals, cremated remains, and perishables.

The recipient must provide appropriate identification.

You are correct the basic insurance that an US customer has won't cover the act of moving your possessions between homes.

The basic insurance that a moving company offers is based on weight, which makes it a terrible option for valuable items because if they lose/break a 5 lb antique, the amount they reimburse will be the same as if they lose 5 lbs of t-shirts. They do have other policies that might make more sense.

Your issue is made even more difficult because of the border crossing making the insurance even more complex, and your concern about duty. You will have to address the duty issue regardless of the method of the item crossing the border. A shipper will have to know how to handle the paperwork, and you will have to declare it.

For the insurance issue, talk to an independent insurance broker/agent, they may have more options available to offer you. Since you will need coverage for a while in Canada, you might start with an insurance company there.