In a discussion with Frederick Law started in a Top Ten Idea the question was asked "How to Link Part Custom Properties to a Cut List".
Although not automatic this can be done without too many extra steps. However, keep in mind that part custom properties apply to just that - the part. So if you do add part custom properties to a weldment cut list, the same information will be repeated in every line of the cut list.
Adding a Part Custom Property into a weldment cut list simply requires adding the standard SolidWorks syntax for custom properties into the Value/Text Expression cell. This syntax is found in the help file under design tables as well as other references. In general, that syntax is $PRP:"your_custom_property_name_here". Unless this has been changed in newer versions of the software, the $PRP: must be in upper case letters.
Once the property is added to the part cut list it is available in the drawing cut list table.
Alternatively, you can add custom properties directly to the drawing cut list table by adding an equation to the cells of a column. If you select the column header the Equation button will be grayed out. However, if you select one or more cells directly the button becomes active. Within the equation popup dialog box you can select the "Custom Properties" button and select from the drop down list. The Equation will be entered into all of the selected cells.