Wait... Without the Moons Khajiit give birth to Bosmer??
We don't know. In fact, http://www.imperial-library.info/content/words-clan-mother-ahnissi-her-favored-daughterindicates that once, Bosmer and Khajiits were the same. And as the Bosmers must have deviated of the Aldmers... We also know that some sorts of khajiits http://www.uesp.net/wiki/Lore:Khajiit#Ohmes (and the most Bosmerish-like Khajiits are those born when the moons are new...)
As one goal of the Thalmor is to unify the mers (elves) and to return to a higher state, they may have tried to transform Khajiits into Bosmer.
-If they were successful, they may have stopped in order to calm Elsweyr's folk, but they will restart later
- If they weren't (kahjiits didn't have children any more, became monstruous, only had Ohmes, who knows?) they decided to retry it later/they were satisfied by the alliance with the Mane.