Do you legally need the suffix? Would it be better to just leave it off completely in both places?
this!
My legal name is John Middle Smith Jr. (not really)
The passport and the driver's license both have the complete name. The airlines, hotels, car rentals credit cards and Concur (work travel system) have John Smith, without the middle name and without the suffix. Leve the middle name and the suffix off. At a minimum, leave the suffix off. The work ID doesn't even have the suffix...