What is the theological justification for the British monarch worshipping in the Church of Scotland, rather than the Scottish Episcopal Church?

Her Majesty has, on occasions, worshipped at churches of many denominations, including Roman Catholic. She attended Vespers at Westminster (RC) Cathedral on St. Andrew's Day, 1995, to mark the centenary of the commencement of its construction.

Her involvement in the Church of Scotland is on a different level from this. She is more than an occasional, or even regular, worshipper. She is a full member of the Church of Scotland, as well as being a full member of the of the Church of England. She is Supreme Governor of the Church of England only.

Lay members of the Church of Scotland are not required to assent or agree with all aspects of Church doctrine, and neither are lay members of the Church of England. So there is nothing inherently contradictory in being a lay member of either or both churches, without necessarily agreeing with either on a whole range of possible doctrinal issues. Both Churches regard Holy Scripture as paramount.

As Queen, she declared herself to be "a faithful Protestant" in February 1952 immediately she returned to London from Kenya, where she became Queen on her father's death. She also swore

I, Elizabeth the Second, by the Grace of God, of Great Britain, Ireland and the British dominions beyond the seas, Queen, Defender of the Faith, do faithfully promise and swear that I shall inviolably maintain and preserve the Settlement of the True Protestant Religion as established by the laws of Scotland ....

Similar promises in relation to England, and a promises to maintain within the United Kingdom the Protestant Religion, were made during her Coronation Service in 1953.

With regard to bishops, as curiousdannii points out, the Church of England does not necessarily believe that the apostolic succession of the episcopacy, via the manual transmission of the laying on of hands, is essential. It does however attach importance to the fact that it has maintained it and does maintain it; and has sought to remove some doubts on this by the involvement in C of E consecrations, since the 1930s, of Old Catholic bishops from the Utrecht Union. This is done without absolutely claiming that it is essential, but in recognition that some people think it is, or might be. If the Queen believes, in common with probably most Anglicans, that it is not essential then the different polity presents no personal problem to her.

The Church of England also teaches that every national or particular Church has the right to ordain its own rites and ceremonies. The Church of Scotland teaches that all Churches are more or less pure.

In fact, the Queen is not the only Church leader who might be accused, wrongly I think, of changing her beliefs according to which country she is in. The Pope, when in Rome, does not accept the canonicity of the 3rd Book of Maccabees, but there are Eastern Catholic Churches which do accept it. The Pope recognises these Eastern Churches as the authentic expression of Christianity within their respective geographical areas. Does he alter his beliefs when crossing the border?

Queen Victoria was the first monarch since the Union of the Crowns, 1603, to spend a significant amount of time in Scotland. She took Holy Communion at Crathie Church, near to her holiday home, Balmoral Castle, in the Aberdeenshire Highlands. This caused something of a furore in some parts of the Church of England, who argued that, as an Anglican, she ought, if she had wanted to take Communion, to have done so in the Scottish Episcopalian Church, the Anglican Church in Scotland. Victoria much preferred the simplicity in the Highland Kirk, to the more elaborate services she was used to in England.

Victoria pointed out that the bishops of the Scottish Episcopal Church were "mere dissenters". For one thing, unlike their English counterparts, they had not been appointed by her. Her view seems to have been that neither the Church of England, nor the Church of Scotland, was perfect, but that each was the authentic, national, brand of Christianity in its own country.

While accepting curiousdannii's argument that there is no theological objection to the Queen worshipping in any Protestant Church, given religious tolerance; I suspect the reason, at least historically, is more to do with recognition of the importance and status of a national Church within its boundaries.

There are many justifications for having multiple denominations, each with its own particular doctrine. But there are also arguments against schism and in favour of unity. At least to Queen Victoria, and I suspect to her great-great-granddaughter, the concept of a national Church, and the idea of unity as a good thing where it can be had (though not to be enforced), was more important than questions of whether or not to have bishops or any of the other things about which Christians may disagree.

The theological justification is that the unity and status of the national Churches is seen as of greater importance than the differences in doctrine.

There's no inherent incompatibility. Both denominations believe in forms of apostolic succession (though not the kind asserted by the Catholic Church.) Wikipedia explains:

The Anglican Communion "has never officially endorsed any one particular theory of the origin of the historic episcopate, its exact relation to the apostolate, and the sense in which it should be thought of as God given, and in fact tolerates a wide variety of views on these points". Its claim to apostolic succession is rooted in the Church of England's evolution as part of the Western Church. Apostolic succession is viewed not so much as conveyed mechanically through an unbroken chain of the laying-on of hands, but as expressing continuity with the unbroken chain of commitment, beliefs and mission starting with the first apostles; and as hence emphasising the enduring yet evolving nature of the Church.

Jus Divinum Regiminis Ecclesiastici (English translation: The Divine Right of Church Government), which was promulgated by Presbyterian clergy in 1646, holds that historic ministerial succession is necessary for legitimate ministerial authority. It states that ministerial succession is conferred by elders through the laying on of hands, in accordance with 1 Timothy 4:14. The Westminster Assembly held that "There is one general church visible" and that "every minister of the word is to be ordained by imposition of hands, and prayer, with fasting, by those preaching presbyters to whom it doth belong".

From the Presbyterian perspective, all Presbyterian elders are actually bishops. Pressies have the most bishops! The biggest difference between Anglicanism and Presbyterianism is that wherever Anglicans have one leader, Presbyterians will have a plurality. This happens at basically all structural levels.

The formal definition of the Presbyterian church structure that we know today was developed by the Westminster Assembly of 1643-1653. The Assembly was called by the English Parliament and was intended to bridge the divisions between puritans, the Church of Scotland (who already had a form of Presbyterianism), and the more traditional Church of England episcopalians. James VI & I had said that he wanted to push episcopalian government on the Church of Scotland, which they, naturally, were not happy with. This conflict actually devolved into a couple of small wars (and arguably resulted in the English Civil Wars and the execution of Charles.) Against the wishes of Charles, the English Parliament called the assembly, and the majority of the assembly were Church of England clergy, though a few Scots were also sent. In return for a military alliance, the Scottish Parliament had required the English Parliament to bring its church government in line with the Scottish Presbyterian model. The majority of the people selected for the Assembly did support Presbyterian governance, and that's the conclusion they reached.

But Cromwell won the civil war before the Assembly could finish its work. His parliament established religious tolerance, and the Westminster Standards were not formally adopted in English. Individual parishes could adopt Presbyterianism, and 64 out of London's 108 did. But when the monarchy was restored the Assembly was repudiated, and laws were passed that returned the Church of England to a strict episcopalian structure. As a result, as many as 2500 ministers left the Church of England.

I think the theological justification is that the Church of England has never seen itself as the only legitimate church. It has always recognised its sister church in Scotland, even before the Westminster Assembly when the initial form of Presbyterian polity was in place. The Church of England has its own theological arguments for episcopalian polity, but it can't deny that Presbyterianism has strong Biblical and theological backing, and that it has a long history in England and could have become the official position of the Church of England had some battles turned out differently. For the British monarch to worship in different denominations is natural given that the English and Scottish crowns are separate. It would also be appropriate for the monarch to worship in other unestablished Protestant denominations in recognition of the centuries old (and hard won) principle of religious nonconformity.