How do I model cartoon eyes for FBX Export?

Hey, I have a model that I am happy with using lattice modifiers and shrinkwrap modifiers for some cartoon eyes (the shrink wrap are for the eye lids). The problem is, I am trying to get this model to the Unreal Engine exporting using FBX. I have spent the last few days trying to figure out how to get my model to export properly and have come to the conclusion that I cannot. So, without using lattices, how can I create some cartoon eyes (think Zelda Windwaker) that maintain their shape while still functioning as eyes (look at an object, blink, etc.)

Here is my existing model in case you wanted that.

Thanks for the advice!

This is the most common way to make eyes. you need a perfect sphere, and can not use something deformed like you have. what you have is an egg shape. that wont work.

what i got here is a perfect sphere, where i cut out the backpart, and flattened the iris, cause that is very common, and gives the eye a better overall look.