The pair and rapper Cowboy Troy will headline the Great America Alliance Inaugural Gala, which is set for 19 January (17) in Washington, D.C. - the day before the U.S. President-elect takes office.
He also won Trump's reality TV series Celebrity Apprentice in 2011.
"We're thrilled to be performing in our nation's capital during this historic moment in time," Rich said in a statement.
Former New York City mayor Rudy Giuliani and actor Jon Voight are also expected to appear at the pre-inauguration event.
Confirmed performers for the actual inauguration on 20 January (17) include the Mormon Tabernacle Choir and young opera singer Jackie Evancho.