For six genera, including the Gambian pouched rat, the CDC banned their sale, or even the removing them from your house.
The question's a tricky one.
I most want mongooses (dwarf and/or banded), but only one species is actually invasive and legally banned (the small Asian mongoose), and I don't want that one of those. The other eighteen aren't banned by the text of the Lacey Act and weren't banned through normal procedure; they're banned because the FWS made a clerical error and doesn't want to admit its mistake. I think that in theory, one could sue the FWS to get them removed from the list. But since public comments and scientific evidence seemed irrelevant when they decided to ban several snakes, there's a risk they'd just relist the non-invasive species of mongoose via proper procedure in order to restore the status quo. In short, you'd likely need money to get the mongoose species I want off the list and influence to keep them off, but it wouldn't be as hard as the other "potentially invasive species."
Flying foxes aren't invasive either, AFAIK. I believe they were banned in the original Lacey Act because they're considered agricultural pests in their native range. They probably come after the mongooses, but again, they're not actually invasive, just banned.
Brushtail possums actually are invasive (though aside from Hawaii, the Marianas, and American Samoa, all with their own restrictions, it's highly unlikely they'd be invasive anywhere in the US), and I kind of want one. But then, there's so many other animals I want more that I doubt that I'd get one even if they were legal, unless they were really common as pets and a lot of people were rehoming them on Craigslist.
In short, I can't decide which is most appropriate for my selection.