> Apparently "Burma" is supposed to be pronounced in a non-rhotic fashion, but would it have killed the non-rhotic speakers to write "Bama"? Then the rhotic speakers who form the majority of English speakers would have at least had a fighting chance to get it right.
To be fair, Bama wouldn't be the right pronunciation. The easiest way for an RP speaker to indicate what the right pronunciation should be is to write "Burma". An American might try Buhma.
And your complain about "rhotic speakers" being "the majority of English speakers" could only have been taken into account of the British transliterator had access to a time machine to determine the future.
At the time, a person in the British empire wouldn't have been considering the opinions of an American, because America and Britain had separate spheres of influence and a generally amicable but not integrated existence. To them, RP was the language of the elite and that was all that mattered.
But if they were to consider the situation in America, they would have seen that in America as in Britain, non-rhotic accents were on the rise and were somewhat prestigious. If they were going to consider the future situation of the 21st century globe and how someone might pronounce the word in an American-centred word, it would have been entirely reasonable for them to have supposed it would have been one with a non-rhotic accent.
Another matter you might consider is that a very strict spelling would not necessarily help matters anyway. The J in Beijing is pronounced more-or-less like an English j, but that doesn't stop people from mutilating it into the s of measure. Or the person who responds to you, who observes that "Punjab" should be pronounced roughly like the English spelling should indicate, but which is often pronounced with the u of push instead - both because of a widespread assumption that all words in foreign languages are pronounced like Italian, Spanish or French.
And the final matter, which might hopefully set your blood at some ease, is to note that when a word is borrowed into a foreign language, it is subject to different pressures. You are not obliged to pronounce "Burma" or "Myanmar" the way any particular natives of Burma or Myanmar do. The English words are English words, intended to help English speakers identify a place in a generally non-offensive manner. There will always be variation between the source pronunciation and the destination pronunciation, especially once a few hundred years have intervened, and that's fine. The question is can we all agree that these are suitable words for referring to that country?
English is one language with multiple dialects, so words that enter the language from a certain dialect will reflect that history to a greater or lesser degree. Even when there isn't an "r" in the spelling, Americans still manage to put an r into the pronunciation, as with Goethe or Gödel. So to blame the British for unetymological r is just false. Americans do it because they recognise the unity of the language, and they're communicating rather than upholding any particular notion of perfection. It is unreasonable to say that the only reason for the British to do the same is that their life depends on it.