How do I prove what I lost to an insurer?

With records made before the loss, because everything made afterwards is reconstruction:

• An asset register with purchase dates, prices and serial numbers.
• Invoices and receipts stored off-site or in the cloud, not in the drawer that burned.
• Photographs of equipment and premises, refreshed occasionally.
• A record of the disruption itself - when you stopped trading and what it cost.

The claim is easier for the same reason the recovery is: because you wrote it down while everything was calm.