Technically, "possum" refers to species of the suborder Phalangeriformes, distributed in Australia. However, "opossums" are also commonly called possums (in America).
So, "Falangero" is correct for "possum", and it's derived from the latin name of the suborder they belong (Phalangeriformes). Anyway, I guess possums are not quite common animals. For "opossum" the spanish word is "zarigüeya". "Zarigüeyas" are common and well known animals in America (only american marsupials). So, it depends on what animal are you really refering to.
Anyway, I think "posum" is totally understandable, and if you check in spanish wikipedia for "posum", you get the correct answer. I just ask to a couple of friends (spanish native speakears too) about the meaning of "Los falangeros viven en Australia" y "Los posums viven en australia". Falangeros in the first sentence isn't uniquely associated with an animal, but posum is. I also think in some contexts, "falangero" could be easily interpreted as "thief" rather than an unknown animal, for people ignoring the real meaning of the word. Again, it's not a common animal.