It's not okay in my mind for this child to be outside without sun screen. Even with clothing that offered more protection, I'd still expect for sunscreen to be applied. I read somewhere a long time ago that it only takes one instance of sunburn in a child under 5 years to create a significantly higher expectation of skin cancer in later life and that has always stuck firmly with me.

For this particular child I think you have to first assess if you've reached your limit or not. Most of us aren't going to judge you if you are there and if dealing with this particular parent has just got to the stage where you are done investing your time, effort and worry in a parent who doesn't appear to even be trying to contribution to a solution.

I personally think it's totally unreasonable for a parent to provide a sunscreen and then complain when it's used, to even consider that it's better for the parent that a child is outside without sunscreen vs the parent having to invest a little effort in bathing their child at the end of the day, for a child who is known to have digestive issues not to have spare clothing, or for a child is skin sensitivity issues to not have diaper cream.

I don't think it's up to you to have solutions to all these issues and I certainly understand that you came here for support and not to be belittled. I don't believe a stash of hats and an investment in business deductible expenses will make you feel that the World is full of unicorns farting rainbows.

I also understand that the objection of you posting wasn't to be told ways to turn your frown upside down but as to see if your requirements are reasonable.

Your requirements are not unreasonable.
You feeling frustrated that this parent is endlessly picking at your service even though most of these issues are of their own doing, is very understandable. You aren't a stepford wife with a fake smile and a pretentious faux outlook, you are a professional carer who sees that the level of care from this parent is lacking and you wouldn't be human not to feel some frustration at all these things.

There absolutely is a point at which it's not worth it. I wouldn't buy car seats for a client because they are tax deductible and you shouldn't have to be put in a situation where you are making purchases specifically for one child whose parents are failing to provide the essentials.

I imagine it's incredibly stressful dealing with this client and likely you are dreading her arrival time to see what crazy situation is coming to your door with this client each morning. The fact that on top of it all, she resents paying would be the icing on the cake for me.

I choose to make my day home space calmer and happier not by aiming to fix my client's issues but by choosing client's who fit well. If you decide to replace her, don't feel like you failed.