The people of Tuvalu are only some 10.000 persons
You are just plain wrong! The population is 11,999 not 10,000!!!
https://www.cia.gov/library/publications/the-world-factbook/geos/tv.html#Intro

- LOL
For other's interested in topic Links about Tuvalu's present problem.
http://www.truehealth.org/climnw05.htmlYou rise here a couple of issues which are not immediately "Gay issues", but I would say they are important for the planning of any future territorial Gay enterprise. On the one side, changing geopolitical realities would increase frictions between states because of the scarce ressources, on the other side every densely populated society must handle its environmental pollution.
It was and is my understanding that reality has to be considered in all of this. Although a Gay Nation will be politically interested in serving the needs of the GLBT, it also has to exist in nature, in the global economy, on a world which will be facing some major changes. Although those changes may be a "bad thing" all around, they can, and should, be twisted around to "good". When life hands you lemons, make lemon-aide - or so my Grandma always said. Climate change and the pressures of economics, limited resources, and social changes may very well open the doors of possibilities where a stagnant "perfect" world has not room for any growth. I doubt I can stress enough the importance for our Gay Nation to take advantage of these things, to learn from the mistakes of the past, to plan not only for a Politically Gay state, but also one which is a safe haven in future.
I strongly feel that although most "first world" GLBT may not want to leave their homes today for a pacific retreat, but when the realities of a changing climate come home to roost they will want to move, especially if that retreat is rising upon the waters and will not suffer as much as the rest of the world.
Another option, which was not considered so far, is the prospective of certain pacific nations to drawn in the ocean, and this pretty soon. The nation of Tuvalu, for example, is seriously contemplating to abandon their little attols, since they will be overflooded in the next future. Now, with the people of Tuvalu having no use of the place anymore, why shouldn't we simply pay them out and "legally" obtaint the submerging reefs?
Actually I thought about this too, perhaps not Tuvalu itself, but I have considered the option of buying atolls/islands (one or more) and building them up to face the rising waters. Atolls particularly crossed my mind. They are natural "harbors" being rings of land or near land with a shallow center - ideal for a protected marina type setting. My mind treads there lightly, I think that the future will see atolls becoming reefs which are getting deeper and deeper. So my thought is why buy land today that isn't going to be there 50 years from now (or sooner)?
However if I was to pick one of these atolls to work with it would be Nukufetau:
From the image above you get a good idea that it is already ideally shaped for reclamation from the seas once it is submerged again as a reef. Roughly rectangular I would assume that beneath a layer of sand there is coral "bed-rock" upon which a "wall" can be built. I also assume that the perimeter reef is shallow, say 25 feet below the water surface (if that). Of course what I am thinking would have Green Peace up in arms and violently protesting, but hey the reefs are being devoured by the starfish population there - but it would be rather "simple" (in relativistic terms) to wall off the inner lagoon from the sea and use that as safe harbor for a floating city/town/etc. It wouldn't have to be a water proof wall, it doesn't even have to be solid, meaning it should have openings to the sea beyond providing for the continual flow of "fresh" water in and out with the tides "flushing" the lagoon.
If people are more industrious then they may dredge up dirt from outside of the ring and "build" land inside the wall.
In the case of Nukafetau we are talking approximately 40 square miles of interior "lagoon" to fill.
Nukafetau is approximately 8 miles by 6 miles according to the line map.
But then the thought comes to mind that 50-100 years from now ocean levels may rise 100 to 200 feet - so much for your island.

A line map (below) shows that it already has some structures, an abandoned airport (I would assume relatively easy to start it up again) and a relatively small population in and of itself.
"an estimated population of 796" (Link below) Infrastructure of a kind is there, I do not know what kind of sewage treatment they have or if they have electricity.
They have a well - (wow) how viable that water source in future will be with constant high tide flooding is an unknown. However there are alternatives and I would assume that initial colonists would work on that ASAP.

Tuvalu Islands:
http://www.janeresture.com/tuvalu/tuvalu.html MAP:
Nanumea: MAP:http://www.janeresture.com/tuvalumaps/nanumea.htm
Nui: MAP:
http://www.janeresture.com/tuvalumaps/nui.htmVaitupu: MAP:
http://www.janeresture.com/tuvalumaps/vaitupu.htmNukufetau: MAP:
http://www.janeresture.com/tuvalumaps/nukufetau.htmFunafuti: MAP:
http://www.janeresture.com/tuvalumaps/funafuti1.htmNukulaelae: MAP:
http://www.janeresture.com/tu8/nukulaelae.htm