I'm guessing TRI is Type Rating Instructor. There's no equivalent rating in the FAA system. To instruct a type rating student in an aircraft you would have to hold a commercial certificate with the type rating and a flight instructor certificate. If you are going to instruct in a sim you would have to meet the experience requirements for an ATP but not necessarily hold an ATP and be on the training center's list of approved instructors.
If you hold an EASA ATPL you would have to have your CAA verify that to the FAA, completed a week long ATP Certificate Training Program, pass the one written test and pass a checkride. If you are doing the ATP and a type rating together you would also have to complete an aircraft training program, both classroom and sim, before the checkride.