Catalans Dragons general manager Alex Chan has also been fined for comments they made about the panel's verdicts.
Carter issued a statement on July 14 following the panel's decision to charge three players from the Wakefield-St Helens Super League match five days earlier.
One of the players, Saints second-row Sione Mata'utia, was forced to miss the Challenge Cup final after being found guilty of a dangerous tackle.
Carter described the citings as "random charges" based on "one nameless person's opinion on a Monday morning".
He also claimed it was futile challenging the decisions because "it's impossible to win an appeal unless you've enough evidence to free Ronnie Biggs".
He was fined £2,000, with £750 suspended until the end of the 2022 season, while Agar was fined £1,000, half of it suspended until end of the 2021 campaign, and Chan was given a £500 fine, suspended until the end of the campaign.
The RFL says all three were found guilty of breaching its operational rules and the Respect Policy but have a right of appeal.
A statement said: "The RFL works constantly with clubs on the efficacy of the game's disciplinary procedures. Those procedures include the right of appeal, and the opportunity for internal feedback."