They do not seem to care about a suffix such as Jr., III etc. I am a "III" and my DL does not have it but my passport does.
Read
TSA's own rules on Secure Fright names. (thanks to the OP for this link)
As noted on page 1 (page 4 of the pdf) you get a first, middle, and last name data field, 35 characters each. Only dash ( - ) and single quote ( ' ) allowed as numeric characters.
"Do not include suffixes (e.g., jr.). Truncate names longer than 35 characters."
The US centric view of the TSA drones is sickening. They assume everyone in the world has a nice US Anglo type name: "John Allen Whitfield." First Middle Last, no punctuation except a (') like "O"Donnell", and better add a (-) for the hyphenated modern marriage names "Whitfield-Jones". But that's it. None of them funny furreign names wanted or allowed.