That someone would say we have nothing to lose is really quite concerning.
If you have an animal in your life whom you love and who means everything to you. Then there is always something to lose. Animals don't rank the same in a lot of peoples minds and they have very little to no legal rights in a lot of societies so some body with the weight of law behind them could quite easily come knocking on your door and take your animal from you. They could then in the best situation be re-homed to somewhere that would at least take care of them, that is of course after they have been desexed to supposedly cleanse them of the sexual stink you put on them through your "sexual depravity'. That is the best outcome the worst outcome would save the authorities a lot of time, by simply putting the animal out of it's misery via a bullet to the brain to save them from the memory of the "perverted" acts a zoo did with them.
Laws can be changed easily enough, giving powers to people to do everything I just mentioned. Would I like zoo to be destigmatised and for more people to understand that interspecies love is quite possible without harm to any party to the arrangement, absolutely. but it has to be a gradual change chipping away at the edges of opinion. Getting in people's face or making grand gestures that are nothing but a flash in the pan isn't going to do anything productive in the long term.
Humans in general don't like things that are different. So even if a fully evolved anthro canine with a degree in quantum mechanics stepped out of his house one day and announced in flawless and clear English that he was in a relationship with a human people would still have a problem with an animal and a human having sex with each other, it's just the dumb ass nature of humans as a species. If ET turned up tomorrow and was covered in fur and looked cute people would fall in love with them, and sure as chickens hatch out of eggs those that were alien lovers would be condemned for it if not hunted down and harmed by those that disapproved.
Different is always seen as bad, till that mindset changes in the majority of peoples minds, "Out of sight out of mind" is the safest way forward.
Opinions stand the best chances of being changed, by simply showing all who meet you and your animal(s), that you care for them deeply and take care of all their needs. This simple act, stands the best chance of reducing the severity of problems should your zoo relationship ever be exposed.