I was thinking about that just now. Based on a study of exactly zero evidence and no reading of authorities whatsoever, I decided that it was impossible.
The Scottish Government can't argue that its own human rights had been breached because as a public body it doesn't have any, and certainly canot itself claim to be a person with a GRC now denied the right to take up a place on the bord of another public body reserved for women. It can't continue to act on behalf of such an individual GRC-holder, because in FWS there isn't one. It also doesn't have a direct interest in equality law since that is not a devolved matter. And - the Scottish government is a statutory creation of the United Kindom as a state. So, ultimately, the UK could simply abolish the Scottish Government to extinguish any action between the two.
And, now I've started to look for authorities, I find this, in Assandize v. Georgia: at 149:
The Court thus emphasises that the higher authorities of the Georgian State are strictly liable under the Convention for the conduct of their subordinates (see Ireland v. the United Kingdom , cited above, p. 64, § 159). It is only the responsibility of the Georgian State itself – not that of a domestic authority or organ – that is in issue before the Court. It is not the Court's role to deal with a multiplicity of national authorities or courts or to examine disputes between institutions or over internal politics.