Being static images, your drawing tables created by Bommer will not update automatically to include any underlying data changes; this is an unfortunate limitation of the process Bommer uses to get bill of materials data onto your drawing sheets. To make the process of recreating your tables a bit easier, Bommer automatically saves the last export configuration for a given drawing. This includes properties, selected components, item number assignments, and output configuration, and is stored in the drawing attributes so that the configuration is saved with your drawings.
To update your drawing table with new data (after using Fusion 360 to update your references), open the command "Export table image..." when editing a drawing for which you've already exported a bill of materials table. Wait for it to load the new data, and then press OK to export the new table. You can also tweak item numbers, change style properties, widths, etc in the same way. Follow the same procedure to place and resize the image (click here for more info) and then save your drawing to save the new table and export configuration.