I cannot find any direct link between Dorothy Arundell, Lady Weston and Henry Boyle (1669–1725) 1st Baron Carleton, the first known owner of the portrait, however. The closest I can find is that his great-grandmother was Lady Frances Cecil (1593 – 1644), while another Lady Frances Cecil (1581 – 1653), the other one's cousin, married Nicholas Tufton, 1st Earl of Thanet. Their daughter Mary married Sir Edward Bishopp, 2nd Baronet (1602 – April 1649), one the descendants of Dorothy Arundell's daughter Jane.
That is not very close.
Searching among the female ancestors of the Baron another possibility is:
ELIZABETH BROOKE (January 12, 1561/2-January 1596/7)
Elizabeth Brooke was the daughter of William Brooke, 10th baron Cobham (November 1, 1527-March 6, 1597) and Frances Newton (1539-October 17, 1592). Although she had a twin sister, Frances, only Elizabeth was christened at court, in the Chapel Royal at Windsor. Her godmothers were the queen and her aunt, Elizabeth Brooke Parr, Lady Northampton. According to David McKeen’s A Memory of Honour: the life of William Brooke, Lord Cobham, she was already at court in January 1581/2 when she received a New Year’s gift from the queen of 6s. 8d.(she gave the queen a ruff) and was one of the gentlewomen of the privy chamber by 1586. Other sources say she first went to court in 1588 and that she immediately captured the affection of Robert Cecil, later earl of Salisbury (1563-1612). In fact, Elizabeth and Cecil probably knew each other as children, since their fathers were close friends. Cecil was concerned that she would reject him because of his spinal deformity. In a letter, he wrote: “The object of mine eye yesternight at supper hath taken so deep impression on my heart that every trifling thought increased my affection. I know your inwardness with all parties to be such, as only it lieth in your person to draw from them whether the mislike of my person be such as it may not be qualified by any other circumstance, with, if it be so, as of likehood it is, I will then lay hand on my mouth.” Apparently Elizabeth was not repelled by his hump. In April 1589 they were betrothed (McKeen says the contract was signed May 31, 1589). She was to have a dowry of £2000 and her jointure would include an estate at Pymmes, Hertfordshire. The death of Cecil’s mother, Mildred, delayed the ceremony, but they were married on August 31, 1589. After that, Elizabeth was often at court. According to All the Queen’s Women: The Changing Place and Perception of Aristocratic Women in Elizabethan England 1558-1620 (1987) by Joan Barbara Greenbaum Goldsmith, she died there. McKeen supplies the information that she died in childbirth. Her children were Frances (1590-1644), Catherine (d.yng), and William (March 1591-1668), although McKeen gives a birthdate of July 1593 for Frances. Elizabeth’s epitaph remembers her as “silent, true and chaste.” Portrait: Elizabeth is included, as a child, in the group portrait of the Cobham Family painted in 1567, although there is some confusion as to which twin is which. In 1590, she commissioned a copy of that painting that included another brother not yet born in 1567. According to David McKeen’s A Memory of Honour: the life of William Brooke, Lord Cobham, there is a portrait at Hatfield of a lady dressed in gold and flowery embroidery of the right style for the 1580s that is usually said to be one of William Cecil’s daughters. He believes, based on the subject’s large nose, a characteristic of the Brooke family, and her small, wry mouth and receding chin, characteristic of the Newtons, that this is a portrait of Elizabeth Brooke Cecil. Susan E. James, in Women’s Voices in Tudor Wills, 1485-1603, makes a case for the portrait at Hatfield House called Anne Morgan, Lady Hunsdon (the same one McKeen refers to?) to be Elizabeth Brooke c.1588-9.
Elizabeth Brooke – A Who's Who of Tudor Women