The myth of Linnaeus being associated is most likely because Linnaeus knew Cators father in law Peter Collinson but did not visit England after his 1735 visit. Cator built Beckenham place house in 1760 and the clock is older because it was brought from nearby Clockhouse built circa 1730 for the Lethieullier family but acquired in 1780 by Joseph Cator, John Cators brother.
The park is on former Foxgrove manor land and not on Beckenham manor land although the latter was acquired by Cator in 1773 and now forms new Beckenham etc.
www.beckenhamplaceparkfriends.org.uk/beckhistintro.html explains