I don't know of a single completely obvious choice, so this is just my thoughts on some possibilities. The order here is just to make this answer less similar to the existing answers.
"Sariga" (or similar: sarigua, sairguea)
I wouldn't go so far as recommending it, but I wanted to mention that another stem used in a few Romance languages' words for opossum is sarig-, as in French sarigue and Italian sariga. (But opossum is also used, and appears to be more common than sariga in Italian.)
The etymology is said to be from Tupi, but the details of what the original Tupi form sounded like and meant are fairly unclear to me. Some Romance forms have [w] (Spanish zarigüeya). Possible Latinized forms could be sarīga, sarīgua, sarīguēa. (I'd go with long ī since the Italian word seems to have penultimate stress.)
Sarigua seems to have had some use as an alternative scientific name at one point (A PRELIMINARY STUDY OF THE NORTH AMERICAN OPOSSUMS OF THE GENUS DIDELPHIS, page 163).
The one advantage I can see is that sarīga or similar would make for a fairly normal-looking feminine noun of the first declension, like formīca, formīcae "ant" or aquila, aquilae "eagle".
A couple of major disadvantages:
Little if any use in Latin.
I found 1 citation of the form "sariguea":
Est memorabile animal Sariguea in nouo orbe, et in India Orientali adinuentum: est catti, seu felis magnitudine; posteriores, ac priores pedes pellicula interposita coniunguntur, velut anseribus in pedibus est pellis, aut qualis est vespertilionibus ...
(Dies Caniculares, Hoc est Colloquia tria et viginti physica, etc)
It's probably recognizable to much fewer people than opossum.
The recognizability is the big attraction of this choice, I think.
As Ben Kovitz mentioned in a comment, it is fairly odd for the name of an animal to belong to the neuter gender in Latin. But as a nominative singular form, opossum is really not compatible with any Latin declension pattern other than second-declension neuter.
"Opossum" is invariable for number in Italian, so treating it as invariable for case and number could be another option to consider. I have the impression that many modern Latin enthusiasts try whenever possible to avoid loaning names as invariable nouns, but it isn't an unheard of strategy historically.
Didelphis (or "didelphys"?)
While not as likely I think as opossum to be immediately recognizable, didelphis is probably fairly easy for a reader to recognize after a little research because of the use of Didelphis as a scientific name. A few languages also seem to have a common noun based on the scientific name: Polish dydelf and the constructed language Esperanto's didelfo.
One thing that makes me a little hesistant to use this, aside from its apparent artificiality as a name, is that it's not that obvious how it ought to be declined.
Nouns ending in -is can have various sorts of stems; native Latin words tend to be third-declension "mixed" i-stems while words borrowed from Greek are often -id- stems. Concretely, this means that the accusative (which is what you would use in the context of a sentence like "Heri [opposum] vidi") would not be didelphis, but would probably be didelphem or didelphidem. (Technically, a "pure" i-stem form didelphim would also be possible, although this was a fairly rare declension pattern.)
Since the names of biological genera are rarely declined, I can't actually tell whether there is an established consensus about the stem of Didelphis. Here is one source that uses "Spiroptera Didelphidis virginianae" and another that uses "Spiroptera Didelphis virginianae".
Unfortunately, I can't use etymology as a guide because I can't find out whether the -is termination of Didelphis is anything more than an arbitrary formation. Wiktionary, in saying that it is from "Ancient Greek δι- (di-, “two”) + δελφύς (delphús, “womb”)", seems to imply that it could be viewed as a variant spelling for Didelphys, but the inflection of that in Latin wouldn't be much simpler, since there were varying degrees of Latinization of words ending in -ys (e.g. the accusative of a word didelphys in Latin might be didelphyn instead of didelphym).