I have experienced the exact same problem as OP with Concur as well as International Flights and spent countless hours with Aadvantage CS trying to get it sorted. I am also a "III" and have had people enter it in same field as last name (as poster above mentioned) with a space, without a space, with capital or lower case letter "I", with numeral "1" and too many assorted variations of the aforementioned to count. The underlying computer systems that all have to communicate with each other often have different formatting fields (ie some have suffix field, some do not) plus any individual entering or searching data can change as well. The best advice I can give you is to have Concur reservations leave the suffix off your name altogether for domestic travel- it will only cause problems.

If you don't already have a Known Traveler Number (Global Entry/TSA Precheck/etc. ), get that ASAP. My understanding is that it usually overrides everything else in the systems. For international flights where your passport (which includes suffix ) is the document that everything has to match, you may have to include the suffix on booking. However, when doing OLCI on AA or Oneworld partner, you still need to exclude the suffix or you will generate a name mismatch and have to check in/retrieve boarding passes at airport after live agent confirms your documents. Since making these adjustments, I have made it a several years and a few hundred thousand miles without further incident (knock on wood).