I'm fairly certain that there has never been a pub called "The Reindeer" in Sydenham or Forest Hill, and probably not in the whole borough of Lewisham. I can't really answer for Dulwich, but I've certainly never heard of a Reindeer there either. Do you have any other clues?
Not much more at all. I'm still on the trail of my g.g.grandfather, and I tracked down some details of a court proceeding from 1889 (selling/letting fish businesses that he didn't own!) where one of the detectives who nabbed him, quoted as finding him in 'The Reindeer Public-House'. He doesn't clarify where, exactly, the place was (of course it may have been no-where near Lewisham!), but I thought it was worth asking, just in case.
Clutching at straws, but I thought i'd try the good-old Sydenham forum.
Paul.
CHARLES GAMER (Detective P). I saw the prisoners together in May in the Reindeer public-house talking to a man named Woods
Lostmind, I've just found the account of the trial on the Old Bailey website; it seems likely that the pub in question is the <Reindeer at 151 Rye Lane, Peckham>.