First off, as noted, Tumi does not have a lifetime warranty. Presumably Tumi is denying your claim because accessories are only covered for the first two years and you are out of the warranty period.
Are you sure the add-a-bag strap was actually part of the original package? The reason I ask is that I bought an Alpha 2 International Expandable a few years ago as a gift and an add-a-bag strap was definitely not included - but the dealer offered a promo that included an add-a-bag strap and the "Accent" set of alternate-color bag tag and zipper pulls and monogram plate along with some other goodies. I can understand that these would be considered accessories and thus not included in the five-year warranty.