Lots of interesting questions with perhaps too many remarkable coincidences.
There must have been quite a number of Henry Grahams born c1860, but between 1850 to 1880 only one was born in Sussex. Furthermore, he was a Henry Leonard Graham (quite unique) who had a mother called Amelia, which dovetails with your research and the Lewes baptism of 1869.
Looking at the broader picture, in 1860, West Hill Street was a reasonably new development on the land around Hodson's mill, so hardly a local back street where you went for a quiet illegitimate birth. I don't think a fly proprietor (cabbie) would employ servants either, Amelia must have just lodged with him and his wife temporarily. As the cabbie informant had no knowledge of her former maiden name he was clearly not related, and the wording on the certificate you have suggests Amelia may indeed have been married to someone called Graham who was lodging elsewhere. At that time, Brighton was also a Garrison town with regiments of foot in the Church Street Barracks and Cavalry in Preston village. Neither offered much in the way of accommodation for the wives of soldiers or married officers. (note 'Esquire' on the Lewes baptism)
I would therefore consider that the baptism of a Henry Leonard Graham (albeit 9 years later) to a Frederic and Amelia Graham (Esquire) of Lewes as a strong possibility, and suggest the reason you cannot find a marriage to go with the birth is that it took place during a posting abroad. (Ireland, India or the Crimea)
OK, it may turn out to be a bum steer, but I think an investigation to verify or refute that line of thinking would be worth undertaking.
Roy G