I tried duplicating the painted Egg Suit data in the JSON file containing the user data for backpacks, but the paint data disappears.

Can't do anything?
I haven't confirmed it, but I'm worried that the paint data might disappear even if I don't duplicate it.
It seems like it might disappear if I reload several times or take the Egg Suit in and out of the backpack.

I'd have to look at the Rust code to confirm, but from the behavior you described, it sounds like Rust is preventing this. There isn't any logic in the plugin that would prohibit having two items with the same sign content.

My initial guess is that Rust is effectively de-duplicating the sign file. If two items have the same sign content, they get the same file. When one of the items is destroyed (e.g., because it was in the backpack and the backpack was unloaded, which destroys the item), the sign file gets deleted, even though another item is referencing the same file.

To circumvent, you could try slightly altering the image on each instance of the item. For example, give yourself one (via backpack or whichever plugin), alter it slightly, then give yourself another of the original, then alter that one slightly, and repeat.