Bauer played youth soccer with GPS New Hampshire Classics Elite, helping them to a pair of state titles in 2014 and 2015. He then went on to play college soccer at the University of New Hampshire between 2017 and 2019. Whilst with the Wildcats, Bauer made 60 appearances, scoring 10 goals and tallying 10 assists.[2]
He was named a United Soccer Coaches First Team All-American in his senior season and was a 2019 MAC Hermann Trophy Semifinalist. He also won accolades such as America East Defender of the Year, ECAC Defender of the Year in 2018 and 2019, and is only the second player ever to be named the America East Tournament Most Outstanding Player twice. He was also a 2018 United Soccer Coaches Second Team All-American and a United Soccer Coaches First Team Scholar Athlete in 2019.[3]
On August 19, 2020, Bauer signed his first professional contract with USL Championship side Birmingham Legion.[7] He made his professional debut on August 29, 2020, appearing as an 84th-minute substitute during a 4–1 win over Charlotte Independence.[8] He was out of contract following the 2020 season to be able to participate in the 2021 MLS SuperDraft.[9]
On January 21, 2021, Bauer was selected 31st overall in the 2021 MLS SuperDraft by Atlanta United.[10]
He officially signed a first team contract on April 15, 2021.[11]
Following the 2021 season, Bauer's contract option was declined by Atlanta.[12] On December 17, 2021, he was selected in Stage One of the 2021 MLS Re-Entry Draft by Nashville SC.[13]