Ultimately not letting them in into new habitats in the first place would be the best solution but that's probably unrealistic

Absolutely! But it can also be very difficult to determine which species that are problematic before we find out the hard way. I would prefer it we completely stopped releasing new species into regions where they don't belong, but it seems like people are very interested in doing this for some reason..

Bounties are typically the best way, but it has to be high enough to be worth the effort. Some species rebound to that as well as their breeding cycle goes into overdrive when their number thin out

Yeah, that is of course true. And then it boils down to whether or not the government can afford or justify using the money on bounties for killing animals instead of improving something else, and we get a whole political discussion going. The best solution would of course if it was possible to force the ones responsible for releasing the animal into the wild to pay for the bounties, but that seems unlikely.