The 25-year-old was shown a red card by referee Mike Russell just 14 minutes into Pools' 3-2 defeat following a foul on Theo Robinson.
Sporting Director Chris Turner had previously said that he would not be contesting the decision, claiming that only the "game's superstars" tended to be successful with appeals.
But in a statement on their official website, the Victoria Park outfit said that "having studied footage of the incident over the weekend the club feels compelled to appeal to the FA".
Should they fail to have the decision overturned, Lee-Barrett will miss the vital clash with Hereford on Saturday.
Turner has already confirmed that Jan Budtz will start if Lee-Barrett is unavailable.