I'm not aware of any 'real' football team/club that played in or was a member of the Scottish Football Association, but they might be worth a call if they could throw any light on it.
A number of books and plays refer to 'Strathclyde Football Club' but only as a ficticious entity (and it saves on court costs than if the writer chose Rangers FC or another 'live' club!). They also featured in an episode of Taggart. There still is a Strathclyde Evangelical Churches Football League, and Strathclyde University have a football club, but none of these are professional.
Good luck!