It's a Spanish word (a neologism, of course, used by a small minority of speakers at present), so pronounce it like any other Spanish word:

https://en.wiktionary.org/wiki/latine#Pronunciation_3

Admittedly the question of how to pronounce a Spanish word in English is non-trivial and subsumes the question of how to pronounce Spanish and the question of how to pronounce English ... but apart from that no special case is required for the word "latine"!