It wasn't fleeting. He was a member for years as a rep.
"Byrd joined with other Southern and border state Democrats to filibuster the Civil Rights Act of 1964, personally filibustering the bill for 14 hours — a move he now says he regrets.[7] Despite an 83 day filibuster in the Senate, both parties in Congress voted overwhelmingly in favor of the Act, and President Johnson signed the bill into law.[8] He also opposed the Voting Rights Act of 1965, but voted for the Civil Rights Act of 1968."