Use a Constituent Query with the same filters
Include your existing query of members in a Constituent Export. Select the desired fields in Output, including Membership > Cards > Name on Card. Be sure to enter an appropriate value (more than 3) in the field "For each Membership, enter the number of Cards to export.
As long as a large enough number is selected, all of the "Name on Card" information will be listed in the export spreadsheet correctly.