Historically you are correct.
According Lavery in his "Arming and Fitting....." on page 62 ff is written:
The programme (of copper sheating) began in May 1779, when it was ordered that all ships of 32 guns and less should be coppered next time they were in dock........ By the end of war in 1783, it was accepted that all ships should be coppered,......
The Cherfull was built in 1806 and slightly sure, that she was also coppered, although she was only in Baltic and Portugal, and I guess otherwise only near Britain in duty
British cutter 'Cheerful' (1806). Dates of service, name changes, previous and next incarnations, dimensions, armament, commanders, officers and crewmen, actions, battles, sources
threedecks.org
View attachment 131242 View attachment 131241
Yes, contemporary models have more or less every time no copper sheating, seems, that the modelers did make this, in order to show better the lines of a vessel and left away this part of a ship.......
I am with you .... the white stuff will look very good on your cutter model