The 1989 Claim of Right cannot be used by the Scottish Government to hold another referendum on independence for the same reasons that the Scottish Government cannot do so under the Scotland Act 1998: they (and by extension, the Scottish Parliament) lack the competence to do so.
This was decided by the UK Supreme Court in Devolution issues under the Scotland Act 1998, Reference by the Lord Advocate (Rev1)  UKSC 31 where the Court held (at 77-82) that a proposed referendum on independence would concern a reserved matter: namely, the Union of Scotland and England. A reserved matter is exclusively within the competence of the United Kingdom Parliament unless it chooses to devolve or delegate some or all of the matter to one of the devolved parliaments.
In this case, Section 29 of the Scotland Act 1998 states:
(1) An Act of the Scottish Parliament is not law so far as any
provision of the Act is outside the legislative competence of the
(2) A provision is outside that competence so far as any of the
following paragraphs apply—
(b) it relates to reserved matters
where a reserved matter is defined in Schedule 5, Part 1, Section 1 of the Scotland Act 1998 as including, amongst other things:
The following aspects of the constitution are reserved matters, that
(b) the Union of the Kingdoms of Scotland and England
In the Devolution (Rev1) case, the Supreme Court held (at 92) that the proposed independence referendum question related to a reserved matter and so was outwith the Scottish Parliament's competence.
The Supreme Court also considered the principle of self-determination as established in international law, which is relevant to the matter at the heart of the Claim of Right. Namely, that the people of Scotland are sovereign and have the right to determine their own independence and so on.
The Court held (at 88-89) that self-determination was expected to occur within the existing framework of the state, per the Supreme Court of Canada in Reference re Secession of Quebec  2 SCR 217 and UN opinion on the matter of Kosovo in Written Proceedings in relation to UN General Assembly Resolution 63/3 (A/RES/63/3) (8 October 2008).
- In its judgment the Supreme Court explained (at paras 136-137) that Canada was a sovereign and independent state conducting itself in compliance with the principle of equal rights and self-determination of peoples and thus possessed of a government representing the whole people belonging to the territory without distinction. It considered that the then current constitutional arrangements within Canada did not place Quebecers in a disadvantaged position within the scope of the international law rule. It continued:
“In summary, the international law right to self-determination only generates, at best, a right to external self-determination in situations of former colonies; where a people is oppressed, as for example under foreign military occupation; or where a definable group is denied meaningful access to government to pursue their political, economic, social and cultural development. In all three situations, the people in question are entitled to a right to external self-determination because they have been denied the ability to exert internally their right to self-determination. Such exceptional circumstances are manifestly inapplicable to Quebec under existing conditions.” (at para 138)
It went on to say that in other circumstances peoples were expected to
achieve self-determination within the framework of their existing
“A state whose government represents the whole of the people or
peoples resident within its territory, on a basis of equality and
without discrimination, and respects the principles of
self-determination in its internal arrangements, is entitled to
maintain its territorial integrity under international law and to have
that territorial integrity recognized by other states. Quebec does not
meet the threshold of a colonial people or an oppressed people, nor
can it be suggested that Quebecers have been denied
meaningful access to government to pursue their political, economic,
cultural and social development. In the circumstances, the National
Assembly, the legislature or the government of Quebec do not enjoy a
right at international law to effect the secession of Quebec from
Canada unilaterally.” (at para 154)
- In our view these observations apply with equal force to the position
of Scotland and the people of Scotland within the United Kingdom. ... The submission went on to state that international law does not, in general, prohibit secession; but the relevant point, in relation to the intervener’s submission based on a right of self-determination under international law, is the absence of recognition of any such right outside the contexts described by the Supreme Court of Canada, none of which applies to Scotland.
On this basis, the Claim of Right 1989 is doomed to fail because any self-determination has to operate within the existing framework of the United Kingdom's political hiercharcy and constitution, and the Supreme Court has made it clear that the Union is a reserved matter, so the United Kingdom Parliament must give explicit permission for any proposed Bill or legal mechanism that affects that.