I didn't want to have large fish because I would hate to lose them to predators. We are close to the Gulf of Mexico and there are a lot of fishing birds in the area. So, I stuck to small tropical fish. In this pond I have Mollies and Guppies and Angel Fish and none of them have bothered the lilies. I did have one year when the leaf cutter ants found the pond and walked out across the lily pads and then ate their way back. They did a lot of damage in one night but I put out bait and killed the nest the next day, it's amazing how much they can carry off in one night.
I had some long finned koi and Wakin in a pond in Calif and luckily they never started eating the lilies or even digging in the pots. I think maybe that was just luck.
It's funny that your fish only like the tropicals. I wonder why.