Brown-Bampoe played
British Universities and Colleges Sport (BUCS) rugby union whilst studying at
Durham University, during which he was called up to the
England rugby sevens team in 2022. He joined
Exeter Chiefs during the 2023-24 season after impressing in BUCS rugby. He spent time on loan that season with
National League 1 side
Plymouth Albion. In August 2024, he scored a hat-trick of tries for Exeter in a pre-season friendly match against local rivals
Cornish Pirates. He also scored in pre-season matches against
Ospreys and
Ulster Rugby, leading to Exeter head coach
Rob Baxter to suggest he was ready for a senior league debut for the club. He made his
Rugby Premiership debut for Exeter on 21 September 2024 against
Leicester Tigers. He rejoined Plymouth Albion in the English National League One on a short-term loan deal in December 2024. He ran in a try from 95-metres on his Exeter Chiefs
European Rugby Champions Cup debut against
Union Bordeaux Bègles on 11 January 2025. In February 2025, he scored four tries in a 43-7 win over Cornish Pirates in the Premiership Rugby Cup. In March 2025, he played in the final of that competition as
Bath Rugby beat Exeter Chiefs 48-14 to win the
2024-25 Premiership Rugby Cup. The following month, he scored two tries as Exeter launched a second half comeback to narrowly lose 24-26 to runaway league leaders Bath on 19 April 2025. ==Style of play==