As I understand it, the case is being run in Scotland on the belief that Scottish constitutional law is more likely to allow this claim than English law is. That is why the case is not being run in the High Court of England and Wales.
The case can't start in the Supreme Court because (apart from specific kinds of disputes) the Supreme Court only hears appeals (see the Constitutional Reform Act 2005). If the Court of Session rejects the case, the plaintiffs might then appeal to the Supreme Court, which would hear the appeal under Scottish law.
Scottish law is different to English law, but there's only one Parliament and one Crown. The letter you linked to goes to great lengths to highlight possible differences between Scottish and English law, but they all occur prior to the Treaty of Union which took effect in 1707. The Scottish Parliament's Union with England Act 1707, section 18, says that Scottish law continues in force except insofar as it is inconsistent with the Treaty. A law which provided for Ministers' or the Crown's powers to be curtailed in Scotland but not in England (in respect to a single indivisible subject matter such as the Parliament) would be inconsistent with the core proposition of the Treaty, which is the creating of a joint Crown and a joint Parliament (sections 1 and 2 of the Act).
The above difficulties are to say nothing of the low likelihood that any court can:
(a) stop Ministers from giving advice, given that the Minister is not exercising any power (only the Queen can actually prorogue Parliament); or
(b) stop the Queen from proroguing Parliament (since Her Majesty is the source of the court's authority, and not subject to it).
(For completeness it should be noted that the Scottish Parliament that passed the Union with England Act 1707 and the current Scottish Parliament are two entirely different things. The current Scottish Parliament was created by the UK Parliament's Scotland Act 1998, and its powers are set out in that Act. The current Scottish Parliament is prohibited from legislating with respect to the union with England (schedule 5 paragraph 1).)
The letter also says that proroguing Parliament would be inconsistent with the European Union (Withdrawal) Act 2018. That Act expressly says that:
- the European Communities Act 1972 will be repealed on exit day (section 1); and
- up until two years after exit day, the Government can make regulations with the force of law to deal with deficiencies in UK law caused by Brexit, e.g. the conferral of powers under UK law on an EU body (presumably the regulations could transfer that power to a UK body) (section 8).
The Act goes on to say that, if it gets Parliament's approval for a withdrawal agreement, the Government can make regulations with the force of law to implement that withdrawal agreement. If there is no withdrawal agreement, then there's no need for the Government to get Parliament's approval.
Not only does the Act not say that Parliament needs to sit in order to consider Brexit, but it expressly sets out a coherent way to deal with the consequences of a no-deal Brexit, namely the regulations made under section 8.
The case is pretty weak and would require a very adventurous judge to approve it.
EDIT: Well fuck me: https://www.bailii.org/uk/cases/UKSC/2019/41.html