An update some ten years later:
Long story short, I have now found a link from my Elizabeth Gollop (& her brother) back to the family of Gollop of Strode.
The first hint at a possible connection to the Gollops of Strode was Eliza Maria Foy. At the beginning of the 18th century, two sons of John Gollop of Strode (mentioned in Burke’s Landed Gentry), John & Thomas, married two Foy sisters, Edith & Mary. The eldest son, John, inherited Strode and Thomas’s family was not recorded in Burkes. However, looking at the Foy family there were two Elizas, one was the wife of John & Thomas’ nephew and the other was the nephew’s daughter.
Unfortunately for me, The Dorset Gollops seemed particularly fond of the names of Thomas and Elizabeth. The upshot was that I could not identify the births of either of “my” Thomas and Elizabeth from the usual sources.
Therefore, I obtained some scanned documents from the Dorset History Centre in respect of Elizabeth's brother Thomas Gollop of Bincombe. None of the documents made any mention of his sister but they did show that he owned a quarry on the Isle of Portland and that he had various financial “issues” with some Weymouth based lawyers. Almost as an aside, he did mention the sale of Lowcombe farm in 1795. I ignored that for a number of years until I found online a copy of John Hutchins’ History of Netherbury & Beminster where it stated that Lowcombe was settled by John Gollop of Strode to his second son, Thomas Gollop of London, from whose grandson, Thomas Gollop of Wyke Regis, it passed to Mr Cox. I used Dorset History Centre’s online catalogue search for any mention of Lowcombe/Luccomb. I ordered some copies of various Deeds that covered the period 1692 to 1793 that concerned various properties, Lowcombe Farm (also Staverlands, Saunder’s Living and Hogbear) in North Bowood. Using those documents, it was then possible to follow the transmission of the property from John Gollop of Strode to Thomas Gollop & Mary Foy to their eldest son Thomas and then to Elizabeth’s brother, Thomas Gollop of Bincombe. The properties were eventually surrendered to cover the various financial liabilities Thomas owed to the Weymouth based lawyers and others.
A sample of the fondness for the name Thomas as per 18th century legalese.
"John Gollop of Dorchester in the County of Dorset Esq son & heir of John Gollop the Elder late of Dorchester aforesaid Mercer deceased, Thomas Gollop the Younger son & heir apparent of the said John Gollop party hereto by Edith his late wife deceased one of the daughters of Walter Foy gent deceased, Thomas Gollop the Elder of the City of London Merchant youngest son of the said John Gollop deceased and Thomas Gollop son & heir apparent of the said Thomas Gollop the Elder party hereto by Mary his wife one of the other daughters of the said Walter Foy Gent..."