Just putting iron's in the fire.
How about making the Falklands Island a major naval base like Gibraltar or even on the scale of Pearl Harbour?
Could that bump up the population in any way especially with all the add on facilities?
It certainly would, but the problem is getting the place to be a naval station of any real note (and that's going to be a lot lower than Gibraltar or Pearl Harbor). The Falklands are strategic in that they sit not far from the Drake Passage (and thus the Clipper Route), and also are a good base for the coast of South America in general. But it begs the question as to why the British just don't grab Tierra del Fuego instead? The Chileans didn't really get to it until the 1850s or so, after all, and the Argentines a bit later. What the British could do if they wanted an even better base in the Drake Passage area is to leverage their economic influence in Chile and Argentina into gaining a significant amount of land in Tierra del Fuego--maybe even exchange the Falklands for the place.
Even with a naval base and some good development, I wouldn't see more than 20-30,000 people there.