This doesn't really answer your question about Sir Stephen Contrell, it is about medals and not coin designs.

From the British Numismatic Society, Laurence Brown's book on British Historic Medals, Ashmolean Museum and my collection.

The twenty-fifth anniversary of the Royal Academy fell on 10 December 1793, and a general meeting was called a week later to reach final agreement on the manner of celebration. It was Farington himself who put the formal proposals to the meeting, namely that they should hold a special Academy dinner; present a loyal address to their patron, King George Ill; and make arrangements to have a medal struck - proposals which were unanimously agreed.

Benjamin West, the Academy president, wasted no time in the matter of the medal design, for within a fortnight of the original meeting he invited Farington to call. 'He showed me', Farington wrote on 30 December 1793, 'a design he had made for the medal. On one side the King's head, on the other Painting, Sculpture and Architecture, represented by three Figures' There is no difficulty in recognizing this as the drawing now preserved in the Ashmolean Museum.

The fact that no trace of the medal has emerged is to my mind conclusive proof of its non-existence.

Many proofs can be found of West's recurrent interest in medals, and in 1815 he took the unusual step of commissioning a medal of himself (Brown 862).

Regarding the coin design letter, Zach adds:

To read the earlier E-Sylum article, see:
BENJAMIN WEST BRITISH COIN LETTER OFFERED(https://www.coinbooks.org/v28/esylum_v28n05a02.html)