Baumann is the son of gentleman driver Oliver Baumann. He made his single-seater debut in the ADAC Formel Masters in 2009, spending two seasons there before joining Heico Motorsport to race in the ADAC GT Masters and the FIA GT3 European Championship. In his first GT3 season Baumann took four podiums en route to fifth in points in ADAC GT Masters, while also taking a series win in the FIA GT3 European Championship at the Slovakia Ring to finish fourth in that standings.
In 2012, Baumann dominated the final season of the FIA GT3 European Championship with Heico Motorsport, taking six victories alongside Maximilian Buhk to claim the title. He moved to Schubert Motorsport and BMW machinery for ADAC GT Masters from 2013, also racing with Lamborghini-affiliated Grasser Racing in the rebranded FIA GT Series. With Schubert he won at Spa and Nürburgring in 2013, took victories at Lausitzring and Slovakia Ring in 2014, and added further wins at Spa, Lausitzring, Sachsenring, and Zandvoort across 2015 to finish as runner-up in that season's ADAC GT Masters.
Baumann joined Mercedes factory team HTP Motorsport for the 2016 season, partnering once more with Maximilian Buhk. Together they scored a win in the Sprint Cup at the Hungaroring and a victory at Silverstone in the Endurance Cup. With points from both series combined, Baumann and Buhk secured the overall 2016 Blancpain GT Series title.
In 2018 Baumann contested a full IMSA SportsCar Championship season with 3GT Racing's Lexus GTD entry alongside Kyle Marcelli, winning at Mid-Ohio and Virginia International Raceway en route to fifth in the class standings. After a reduced programme in 2019, he returned to Mercedes-based teams in 2020 with SPS Automotive Performance in the GT World Challenge Europe Sprint Cup Pro-Am class. He remained with SPS into 2022, taking three class wins during the Endurance Cup season including a notable appearance at the 24 Hours of Spa alongside Kenny Habul.
Baumann stepped back to occasional appearances in 2023, contesting the Gulf 12 Hours for SunEnergy1 Racing and winning Race 2 at the Barcelona round of International GT Open. He returned to full-time competition in 2024 with SPS in International GT Open and with AlManar by GetSpeed in the GT World Challenge Europe Endurance Cup, taking a Gold Cup class win at the 24 Hours of Spa with Mikaël Grenier, Al Faisal Al Zubair, and Philip Ellis.
In 2025, Baumann partnered Valentin Pierburg at SPS Automotive Performance in the Pro-Am class of the GT World Challenge Europe Sprint Cup, winning class victories at Algarve and Le Castellet and amassing seven further podiums to clinch the Pro-Am class title at the Monza finale.
Baumann's career combines factory-level success with enduring competitiveness through the Pro-Am ranks. His dual titles with Buhk across the GT3 European and Blancpain series, plus a Pro-Am championship in 2025, mark him as one of the most consistent Austrian GT racers of his generation across a career now spanning more than fifteen years.