John Ayto, Oxford Dictionary of English Idioms, third edition (2009) includes a brief discussion of "happy as Larry" under a primary entry for "happy as a sandboy":
happy as a sandboy extremely happy ; perfectly contented with your situation.
An 1823 dictionary describes a sandboy as an urchin who sold sand in the streets, and according to the same source the expression jolly as a sandboy was already proverbial by that date for 'a merry fellow who has tasted a drop'. A common British version of the phrase is happy as Larry, Larry being a pet name for Lawrence. This saying is sometimes connected with the renowned boxer Larry Foley (1847–1917); on the other hand, it may owe something to larry, a dialect word used by Thomas Hardy, meaning 'a state of excitement'. The North American version is happy as a clam, which apparently originated in the early 19th century on the east coast, where clams are plentiful: the full version happy as a clam at high water explains the source of the clam's satisfaction.
The only instance of larry that I've been able to find in British regional glossaries of the nineteenth century is from Georgina Jackson, Shropshire Word-Book (1879):
LARRY [laar'-i'], sb. a confused noise, as of a number of people all talking talking together.—Pulverbatch. 'I 'eärd a fine larry las' night—folks gweïn down the Moat lane.
But that same word-book has an interesting definition for Larrance as well:
LARRANCE [laar'-uns], sb., var., pr. the genius of idle people. They are said to have Larrance on their back. Com. 'That chap's got Larrance on 'is back, 'e dunna do 'afe a nour's work in a day.'
So might "happy as Larry" mean "happy as the genius of idle people [that is, 'happy as Larrance']"? I don't know, but it sounds pretty happy.