One, slightly sarcastically meant, path to Flensburg as West German capital.
Lauritz Lauritzen was traffic minister in the 1970s and the only German Minister of Traffic to dare introduce a speed limit on German Autobahnen.
Maybe, as the oil crisis is relieved, Lauritzen does not bow to the demands of ADAC and the car industry, until the latter instigates a "coup". The CDU votes out Chancellor Brandt with a vote-of-no-confidence and the car industry, together with the Kraftfahrt-Bundesamt which is located in Flensburg, makes it the de facto capital of Germany. By the 1990s, the German version of the Business Plot, centred around the KBA in Flensburg, has gained so much power in the Bundestag that the capital is de jure moved to Flensburg.
Though we would still need to explain why they didn't choose Stuttgart...