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The problem is there is no real reference datum.
Centre yes, but that is also going to change as a result of motion of the rock in the core. The orbit of the satellites will change as a result, and thus the height data is not going to be an exact reproducible measurement. All the measurements will be relative, just choose a reference datum for it.
If you look at the earth, its gravitational field is not symmetrical. It is actually asymmetrical. The asymmetry allows the ocean to rise in some places and sink on other places. Below is a representation of the earth as a function of surface gravity.
I heard a radio program last night on how rising sea levels due to global warming had drowned significant areas of he Solomon Islands. Shock, horror, doom, loss of traditional and indigenous etcetera. No figures were given but there was a strong implication that the mean sea level had risen at least 60 cm in 50 years. Now AFAIK all the oceans are connected, and water flows. So if the South Pacific has risen by two feet in a lifetime, so has the Mediterranean (much easier to measure as it is almost tideless). Has it? I think not.There was just one tiny hint of a flaw in the strongly implied Global Warming is Killing Everything hypothesis. The reporter mentioned that the Solomons are volcanic islands. Yep, right on the Pacific ridge. Here today and gone (geologically speaking) tomorrow.
We can't do a lot about the height of the Solomon Islands. (A fairly cheap GPS system would answer that actual question you pose but I presume that you don't really care about that, since you seem to be seeking to make some sort of rhetorical point)But we can stop making things worse for them by not raising global temperatures and thereby melting the antarctic ice.So, lets try and focus on that.
Plenty could indeed be done about the Solomon Islands ...but I don't see any practical means of preventing volcanic islands from disappearing almost as fast as they appear.
the height data is not going to be an exact reproducible measurement
There is one thing to consider - and I regret that I cannot locate the reference this precise second - but the huge mass of ice aggregated over Antarctica, which is not floating and hence not displacing any water - exerts a considerable gravitational effect on surrounding seawater, meaning that there is a circumpolar water bulge; if the ice melts then that water redistributes, causing a greater rise in sea level than just the ice melting alone.