Anton:
 > The natives heard `Gilbert' as /kirbat/ ...
 
I like this. My original source implied a direct link from "Gilberts" 
(as in "the Gilberts") => /kiribas/, but "Gilbert" (as in "Gilbert 
Islands") => /kiribat/ followed by a pronunciation shift is neat.
But Christmas Island may be a counter-example to your proposal. The 
Gilbertese name is Kiritimati (= /kirismas/). I've always assumed the 
island is so named because it was sighted on Christmas Eve 1777 by 
Captain Cook. In which case there never was a /t/ sound in the 
original name, and therefore the present pronunciation can't have 
arisen by drift from a previous form using /ti/ at the time the 
language was first written down.
Dallen has just mildly rocked my apple-cart by suggesting that our 
English name "Christmas" is just a close rendering of the original 
Gilbertese name. But even if that were the case, it still implies the 
natives were using an /s/ sound when first contacted, rather than 
using /ti/.
Interestingly, the FIPS 10-4 digraph for the *Australian* Christmas 
Island is "KT" - perhaps someone confused Australian Christmas with 
Kiribatian Kiritimati?
Grant