“Mexico” is one of only two cases where I don’t follow the “correct” pronunciation as often as possible (the other being ‘axolotl’, because if I pronounce it with the correct Nahuatl ‘sh’ sound, nobody knows what I’m talking about). If the topic is brought up, I wait to see how someone else says it. If I’m speaking to my largely-latina students, many of whom are from mexico, then I’m obviously going to pronounce it as an ‘h’/‘j’. However, many of them also pronounce it with an x when speaking in English, so I just tend to go with ‘correct’ unless one of my interlocutors says it the american way first. I don’t feel particularly bad about this, since the word “Mexico” also comes from Nahuatl, but nobody actually pronounces it the original way in mexico, so I go with whatever my interlocutor goes with first. (For the same reason I’m not going to call Germany “Deutschland” unless I’m speaking to someone whom I know to be German.)

In general, I try to pronounce loan words the correct way in their mother tongue, whether they be Maori, Xhosa, or French. And yes, I know this makes me sound like a pretentious dickwad when I say "Kwah-sahn’ ", rather than “cruh-sahnt”, and I’ll take sounding like a pretentious dickwad over giving in to my American exceptionalism any day.