One or two points, rather than a thorough review

"After taking Liddesdale they bypassed Carlisle in exchange for a large indemnity.." As far as I recall from putting together an article on Liddel Strength, the Scots already held Liddesdale, but the English held Liddel Strength (on the Esk at the mouth of the Liddel), which David (on the advice of the lord of Liddesdale, who was one of his retinue) besieged and took after a delay of 3-4 days. A quick check against Sumption and 'Lanercost' doesn't seem to throw up any mention of Liddesdale more generally being taken.

"According to Lynda Rollason, the Black Rood, a piece of the True Cross previously belonging to the former Queen of Scotland, St. Margaret was taken from David II and donated to the shrine of St. Cuthbert.^[1]"

Checking in the supplementary volume (vol v) to 'Bain' (Simpson, G G; Galbraith, J D (eds.). Calendar of Documents Relating to Scotland. Vol. Volume V (Supplementary). Edinburgh: Scottish Record Office. ((cite book)): |volume= has extra text (help)) (no date given on title page, but from content of preface published in 1969 or 1970) on the unpreparedness of Carlisle in 1347 (items 802, 803), I fortuitously noted that it records (item 800, page 268) a Treasury memorandum of Jan 7 1346 that on this date the Black-rood of Scotland was taken from the Tower of London and delivered to Walter de Wetewang, keeper of the wardrobe, to be kept by the king's side. The original Latin can be found in Palgrave ... (The previous item (799) in 'Bain' vol v is a memo of April 12 1344 of various relics handed over to a new treasurer by his predecessor; they include a reliquary with some bones of St Margaret of Scotland and 'a silver-gilt cross with a part in the middle of black wood' : it would be OR to take the latter to be the Black-rood, but it is a highly tempting conclusion )

"References" Maxwell's edition of the Chronicle of Lanercost Priory

Maxwell, Herbert, ed. (1913). The Chronicle of Lanercost, 1272–1346. Glasgow: J. Maclehose and sons. p. 330. OCLC 457526322.

is accessible on archive.org, and should perhaps be linked, unless there is some stylistic objection

Maxwell, Herbert, ed. (1913). The Chronicle of Lanercost, 1272-1346 : Translated, with notes. Glasgow: J. Maclehose and sons. p. 330. OCLC 457526322.