English is not really that tricky a language to learn for non-native speakers as a general rule - obviously it might be more difficult for someone from a totally different language family than for, say, a Danish speaker, but most languages have grammar rules that are much more complex. True, the pronunciation doesn't seem to follow any rules but I'm pretty sure that there are many other languages where that's also the case.
Latin (and Ancient Greek) was or is still part of the humanistic educational canon and is still taught at schools throughout the Western World because of that. The Federation seems to me to be the embodient of the humanistic ideals so it makes a certain sense that Latin would also be taught at Starfleet Academy to some extent.
Star Trek is very Anglo-/American-/European- centric while our future will probably be different. I assume that the problem the OP has with the teaching of Latin at the Academy is really that in Star Trek Earth's future is depicted as being dominated by WASPs, basically. That's something that also bugs me occasionally but it's just a TV franchise and not a projection of the future.
I don't know what the storyline of the game actually is but as a compromise, why not assume that Latin is an elective course out of many courses students can choose from and Picard knew that Wesley had studied it?