I think you are all being a little unfair on the sister.
This is her brother and they were close. She will want her husband with her at the funeral as someone to lean on.
She and her husband sound self employed. He goes off presenting and she works from home. They will be small contractors and if it endangers a contract, I can see why she would ask you to avoid some days. They have commitments and you don't know what their financial situation is and unlike employed people they will not get paid for compassionate leave.
From what the OP said herself, the sister didn't actually give any dates initially although said there would be awkward ones but she only provided those details when asked, so that didn't sound like a priority for her, it was the OP. The OP asked and then is offended when she got a reply. Of course it would be easier for the OP to not have to consider these dates but she offered to try accommodate, no one said her sister wouldn't come if she couldn't. People are reading into this, things which haven't been said or done.
We all deal with our grief in our own way, The focus of that grief is normally on the wife and child as its their life he is missing from every single day and its especially hard for the OP and child as she has to keep her wits about her because she has to adapt to a life without her Husband/Father of her child and the resulting financial challenges.
But this sister has lost her only sibling, he wasn't with her in her everyday life, so there is no specific reminder in her daily routine as there is for you, just when her thoughts come around to him. Its a different experience but no less valid.
I think OP, you emotions are just running high at the moment because you have a lot on your plate. You have kindly accommodated your Sil and that was a generous thing to do. But please remember you are not the only one hurting, he has been in her life longer than yours, and whilst its not a competition, she will be struggling with this too and as the wife, you have all the control and there is nothing she can do.
I don't mean this unkindly, I just think that in the times of great loss emotions run high and things get said that cannot be unsaid.
I am so sorry for your loss and I wish you all the best.