My issue isn't that you are having trouble with this child, and are trying to find a good way to deal with it. It's with the fact that you identify the fact that a parent decides to pick his/her son up as a "problem", and believing that an 18 month old is coddled. If this child were older, then I would agree that coddling may be an issue. But an 18 month old? Nope. He has only been in this world for 18 months, and maybe the world is harder for him to adapt to. Yes, as care providers, you have every right to have policies and procedures in place in your own homes, in order to make the day run smoothly for everyone. I am not unrealistic. I don't expect care providers to hold my daughter all day long. I expect them to have standards and policies that benefit the group as a whole. Let's not make the very illogical, uninformed leap from me saying "understand where the kid is coming from" to "that child will be ruined because you pander to his every whim." That is a completely false equivalency, it is reductive, and it distorts the point that I am making. These parents have every right to talk to their son how they want, to feed him what they want, to hold and snuggle and care for him how they want to. If it makes your life more difficult, then maybe you aren't a good fit. But it is extremely unfair to come here and criticize their parenting, when they have no recourse to defend themselves.
At this point in time, this child doesn't seem to be a good "fit". But that may very well change in a few months. I take issue with the fact that kids who have high emotional needs are automatically lumped into the "coddled" group. He is obviously having a very rough time. My friend's daughter was the same way, and it was only recently, after being in care for 8 months, that she has finally come around. Luckily, she had a wonderful care provider who stuck it out. So for some kids, it takes much more time. It doesn't mean that they are coddled, or spoiled, or difficult. They just have different needs, and that is ok. This is what you sign up for when you have a home daycare. Different kids, different parents, different styles of parenting.