I want to see the list and find out how much it costs to maintain and enforce it before I make a judgment call, but whitelist (and an appropriate appeals process) should be the type of appropriate answer in an ideal world.
The worst invasive species (bugs mainly ... but also fish) can’t be controlled through this sort of measure, they can hitch rides on produce and in ballast tanks of ships, and a lot of invasive species that get out are food animals imported under the guise of being pets. The amount of effort it would take to enforce it at borders is also burdensome, how are you going to know that the baby fish in some tank are snake heads and not some other species?
The Dept will add animals to the list that inspectors can be trained to identify. All others will be assumed to be invasive and will be banned. In 3 years they will begin the evaluation process to add to the list of acceptable species.