There is a tag "scarytoprey" or something like that. Once applied to a mob, mobs that fall under the category of prey won't run away. I don't know currently how to single out rabbits only, you'll have to look what makes mobs "preys" and make a custom tag or something.
If you want them to approach the player, you'll have to look into the brain, but I'm not experienced with it.
To make them immortal, I think you can simply remove the combat component or whatever the name was. For the rabbit "item", not sure.