Barter began karting at the age of six and progressed steadily through Australian competition. He finished second in the Australian Kart Championship in 2019 and won the opening round of the 2020 season before that championship was abandoned due to the COVID-19 pandemic. He also competed at the Rotax Max Challenge Grand Finals on two occasions, finishing eighth in 2016 and ninth in 2019.
Barter made his car racing debut in the 2021 French F4 Championship with the FFSA Academy. He was immediately competitive: a victory at Nogaro on Sunday of the opening round was followed by a triple podium at Magny-Cours. Despite a quieter middle section of the season, strong form at the finale — including a win in the final race after a championship rival's disqualification — left him second in the standings with 213 points and two race victories.
For 2022 Barter opted to remain in Formula 4, competing simultaneously in French F4 and the F4 Spanish Championship with Campos Racing. He cited his need to develop qualifying pace as a reason for staying in the category. The French F4 campaign was dominant: ten wins including victories at Nogaro, Pau, Magny-Cours, Spa-Francorchamps, Lédenon, and Valencia, along with seven pole positions and 12 fastest laps. A 32-point deficit to Alessandro Giusti going into the finale — accumulated because Barter was ineligible to score points at Spa and Valencia circuits already used in his Spanish campaign — ultimately cost him the title; he finished second with 241 points. In Spain, Barter won the season opener at Portimão and added victories at Aragón and a hat-trick at Navarra, but Nikola Tsolov dominated the championship. Barter again finished second with 287 points. The 2022 Spanish campaign brought six wins, four pole positions, and 13 podiums from 21 races.
After testing with Campos Racing at the 2022 FIA Formula 3 post-season test, Barter signed with the team for the full 2023 season. The step to Formula 3 proved difficult: he was scoreless for the first four rounds before eighth in the Spielberg feature race broke his duck. He then scored sixth in the Silverstone sprint and sixth in the Spa sprint from reverse-pole. Shortly before the Monza finale, unspecified issues forced him out of the final round and he was replaced by Joshua Dufek. He ended the season nineteenth with 14 points.
In April 2023 Barter participated in the Formula E Berlin rookie test with Maserati MSG Racing. He returned to the Berlin rookie test in July 2025, this time for Lola Yamaha ABT. An appearance during the 2026 Miami ePrix and his subsequent announcement as the team's test and simulator driver followed, with a further planned appearance at the Madrid ePrix.
In 2024 Barter raced in the F4 Indian Championship with Godspeed Kochi, scoring three wins and finishing fifth. For 2025 he competed in the Ultimate Cup European Series Hoosier Formula Cup with Winfield Racing, claiming nine wins and finishing fourth in the standings with 296 points. He also races for Drago Racing Esports Team in iRacing.