If you ordered all the world's diets from least harmful to most harmful with regard to animals, the vegetarian and vegan diets would be towards the less harmful end. But there are many cases where a diet involving animal products is going to involve less animal suffering than another diet that doesn't. For example, compare eating crops grown with saturation pollination to eating honey. Honey is an animal product, and the bees do suffer when you take the honey from them, but it looks to me like saturation pollination is much worse. With most kinds of honey the bees are mostly free to do whatever they want, eat wherever they please, and come back to the hive on their own. The beekeepers want to make sure there's enough nectar locally to support the bees so they avoid overcrowding. With saturation pollination, however, large numbers of bees are released in a small area to maximize pollination and crop yields. Yet because this doesn't involve eating animals or animal products it falls within the bounds of 'vegan'.

What would a diet that was designed to minimize animal suffering look like? Has anyone worked on this?

(I suspect this is a very hard problem. For example, let's say production of a certain food generally entails a large amount of rainforest destruction. This is clearly bad for the animals that live there, but its main effect is on animals that don't exist yet, the ones that won't be able to live there in the future because the land has been converted from fertile productive rainforest to much less productive crop land or even parking lots. If you think that animals in the wild generally have good lives then this is a bad thing, but if you think the typical life of a wild animal is mostly suffering then this habitat destruction is probably beneficial on balance and should be counted in the favor of the food in question.)