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[Another, abridged edition of this work was published in 1899 and is also available at the Internet Archive and Google Books, but it does not have some of the supplemental materials discussed here.]
Harrison does not itemize the children of Sir Henry Weston, but he does quote the will of Sir Henry, in which his daughter Jane is named as his sole executrix. Presumably this is the Jane who married Sir Thomas Bishopp.
Many thanks for the link to the Frederic Harrison book. Harrison says, on p. 163, when itemizing the Sutton portraits, "Portrait of a young lady, dated 1610, aetat. eighteen. This may be Jane Weston, daughter of Sir Richard Weston the second, and sister of the agriculturist. She married Sir Thomas Bishop, Bart.":
So Harrison agrees with CB here about Jane Bishopp being the daughter of Sir Richard Weston.
As John pointed out, the book contains a copy of Sir Henry Weston's will, dated 16 Nov. 1588, in which he refers to her as "Jane Weston my daughter" several times, and appoints her sole executor. It seems far more likely from the context of his will that Sir Henry is referring to his blood daughter Jane, and not his daughter-in-law Jane, wife of Sir Richard Weston. The will was proved 6 May 1592 by Thomas Lovell notary public and "Jane filie dci defuncti Et exequutricis". Since we aren't given Jane's surname, we cannot tell from the probate whether or not she was married in 1592:
The only two children whom Sir Henry mentions are his son Richard and his daughter Jane. So if he had a daughter Elizabeth, as per the Weston pedigree from the 1623 Visitation of Surrey, she died before 1588:
I checked Richard K. Evans's 2007 book 'The Ancestry of Diana Princess of Wales for Twelve Generations'. He states, on p. 306,
"SIR THOMAS BISHOPP, 1ST BARONET, M.P. 1584, 1586 and 1610-11, Sheriff of Surrey and Sussex 1584/85 and 1601-2, born in 1553 and died before 14 Feb. 1626/27. He married, 1st, 19 Sept. 1577, Anne Cromer, daughter of William Cromer and Margaret Kempe. He married, 2nd, at Findon, Sussex 7 Aug. 1600, 5793. JANE WESTON", and the only two sources he cites for the couple are "The Complete Baronetage, 1: 156; The History of Parliament 1558-1603, 1: 439".
But neither of those two sources has such a specific date of marriage for Sir Thomas Bishopp and Jane Weston. CB doesn't provide any date, while Sir Thomas's HOP bio says "before 1589". Sir Thomas's most recent HOP bio, in the 1604-1629 series, published in 2010, says they were married "c.1589":
Evans must have gotten his date of marriage for the couple from the IGI, and indeed "Thomas Byshope" and "Jeen Westone" were married in Findon, Sussex on 7 August 1600:
This marriage date actually fits much better with the birthdate of their son Sir Edward Bishopp, 2nd Baronet, who, according to his HOP 1604-1629 bio, was born on 6 September 1602:
A 1600 marriage date makes it possible chronologically for Jane Bishopp to have been the daughter of Sir Richard Weston & Jane Dister. If she was born in 1584, the year after their marriage, she would've been age 16 when she married Sir Thomas Bishopp.
But we know from the IPM of Sir Thomas that his wife Jane was the daughter of Sir Henry Weston. Sir Henry Weston and Dorothy Arundell were married in 1559. Their two sons Henry (died young) and Richard were born in 1561 and 1564 respectively, and baptized at St Peter & St Paul Church, West Clandon, Surrey:
Jane must've been baptized elsewhere, and so no doubt was born after her brothers. I had originally estimated her birthdate as about 1560, but I'm now revising it to circa 1567, assuming she had to be at least age 21 when her father appointed her sole executor of his will in November 1588. It puts her in her thirties in 1600-02 when she married and bore her son the second baronet, rather late for a lady's first marriage but still biologically possible.
We don't have a date of death for Dorothy Arundell, Jane's mother. Harrison in 1893, unaware of Sir Henry's second marriage to Elizabeth Lovell Repps, assumed that the unnamed wife Sir Henry leaves bequests to in his will was Dorothy, but in actuality it had to have been Elizabeth, whose first husband Henry Repps died 10 October 1566. Harrison includes a portrait of Dorothy Arundell, and states (p. 92), "The date of the picture was about 1575, when she was about thirty-five":
A full-color version of the portrait is here, with the first name erroneously 'Margaret', as in the Weston pedigree from the 1623 Visitation of Surrey. As Harrison points out, the baptism entries for her sons at West Clandon prove the Arundell family pedigrees correct in that her first name was 'Dorothy', not 'Margaret':
At any rate, it appears Dorothy Arundell Weston lived to at least 1575.
As Jane Weston Bishopp is a 12th-generation ancestor of Diana, Princess of Wales, we don't know where Richard K. Evans would've fallen on her parentage: Sir Richard Weston & Jane Dister as in his one source Complete Baronetage, or Sir Henry Weston & Dorothy Arundell as in his other source HOP, which appears to be the correct parents.