*Club domestic league appearances and goals, correct as of 7 December 2024 (UTC) ‡ National team caps and goals, correct as of 22:12, 6 February 2023 (UTC)
Bischof began his career with SC Göfis.[2] He moved to Rheindorf Altach in 2012, before joining the youth academy of AKA Vorarlberg in 2016, where he progressed through all age groups.[2] In September 2020, he signed a professional contract with his parent club Rheindorf Altach.[3] He joined their professional squad for the 2021–22 season, and made his Austrian Bundesliga debut on 24 July 2021, when he came off the bench for Atdhe Nuhiu in the 63rd minute on the first matchday of the season against LASK, which ended in a 1–0 home loss.[4] He scored his first senior goal on 24 October 2021, the 1–0 winner against LASK from an assist by Nosa Iyobosa Edokpolor.[5]
On 18 January 2024, Bischof signed for Austrian Bundesliga club Rapid Wien on a two-and-a-half-year contract.[6] Upon his transfer to the club, he immediately joined Austrian 2. Liga club First Vienna on loan until the end of the season.[7]