Yes, the map was a key (thank you, sillgen - do you have a date for the map?).
I too would like to be sure which building is the original Barnets Place. I think it must be the largest building closest to the road, to the west of the nursery. The eastern end looks, from the google earth image, to have a large multistory section with a hipped roof which would match the description in the 1811 advertisement and the reconstruction suggested by artifis. I am sure it has been subsequently extended and some of the original outbuildings incorporated into the main structure.
A question about the 1811 description - would the "and on the second four servants rooms" mean we are looking for a three story house; or would the servants rooms have been in an attic? I think the house must have been built by Fasham Nairn when he moved to West Hoathly in the late 1770s, early 1780s. In an English context that would be "modern-built".
The change of name from Barnets Place to Courtlands can probably be narrowed down to between 1864 when William Aveling Nairn died and 1868 when a poster advertises the sale of The Courtlands Estate by Norton, Trist , Watney & Co. (document 0253 in the West Hoathly Archive)
