The 360 terminus is at the rear of the RAH, on Prince Consort Road. Looking at the Central London Bus Guide, it's not clear where the stop at South Kensington is. (It looks as if the bus passes, through the one-way system, just to the west of the station).
An alternative would be to leave the District Line at Victoria and proceed to Victoria Bus Station (which is directly outside the rail station). Service 52 starts from there (with the destination shown as 'Willesden'). The route passes directly in front of the RAH.
Chris