I have emailed Greg about this query as follows:
Your query was passed on to me as the expert on mining in the Matlock area.
I concur that the portrait is almost certainly of Anthony Tissington. I have a definitive book on the Matlock mines (supposedly my PhD thesis - over 5,000 source notes) almost ready for publication and from the draft I extract the following about Anthony Tissington in case it may assist your research.
The most influential of Derbyshire mine agents were of the Tissington family from the important mining area of Winster. In 1767, Erasmus Darwin, a prominent member of the Lunar Society, described Anthony and his brother, George Tissington, as ‘subterranean Genii’. Born at Darley Dale in 1705, Anthony Tissington was an erudite mine agent and owner who was barmaster of Matlock Liberty c.1730-35. He seems to have been a brother-in-law of a later Matlock barmaster, Anthony Wragg. Anthony Tissington left Matlock in about 1735, becoming wealthy from coal mining at Swanwick rather than any Derbyshire lead mining investments. When elected a Fellow of the Royal Society in 1767, he was described as ‘a gentleman of great merit and well acquainted with philosophy’. Anthony Tissington leased metal mining rights in the Lake District in 1757, and at Leadhills, Scotland. He also acquired interests from Thomas Chambers of Derby who had made a vast fortune out of the copper trade. It has been claimed that Anthony Tissington ‘was Britain’s richest mining entrepreneur in 1760’.
Sources:
Uglow, Jenny, 2002. The Lunar Men. p144.
Flindall, Roger, 2000. What the Papers Said: Derbyshire in Nottingham Newspapers 1714-1776. p48.
Arkwright Papers, Willersley Castle A/05 No 29. I am obliged to Jim Rieuwerts for this reference.
http://www.royalsoc.ac.uk/library 27 January 2004.
Fell, Alfred, 1908. The Early Iron Industry of Furness and District. 464pp.
Payne, Peter Lester, 1967. Studies In Scottish Business History. 435pp. p118.
Craven, Maxwell, Derbeian’s Diary, Derby Evening Telegraph 9 Jan 2003 pp22-3.
Craven, Maxwell, 2004, in Derby Telegraph 23 Apr 2004