Ok perfectcstorm who is going to supervise this boy. His parents sound as effective as a chocolate teapot, it is not up to the op as she will be very busy with her dc and party. Yes dd might see how it goes, but if its still the same in say a months time than be any means she should not habpve him there
Piglet, it might aid the discussion somewhat if you troubled to read people's posts properly. I will cut and paste from my original post in answer to your question, as there seems little point rewording what I said there; it answers your questions and points in full.
I said:
I think the only solutions are on the one hand not to invite that specific child, and on the other to ensure an adult member of your family who is good (plays well with them, but can be extremely firm) to "mark" that child throughout, in case his parents don't... What I absolutely would not do would be to invite a kid who could not be independently supervised, and turn my child's party into a nightmare for them. If I hadn't a spare adult play partner then the child would not be asked.
I say that as someone whose kid is also 4 this term coming, and is having a joint birthday with another child so both families can afford to invite the whole class. I'm usually very opposed to excluding even difficult kids. But in this instance, it risks your child spending her own birthday party scared and watchful. Not on at all to place his feelings above hers.
I clearly stated that if someone who could do that could be identified (no problem in my own family) and the DD was comforted by that solution, then that might be the way to go - but if no such person could be found, then no invite, because her DD's feelings had to take precedence. What, precisely, is your problem with that suggestion and statement?
I sometimes think people get so determined to argue their own points over and over on AIBU that they completely overlook what others are actually saying, just so they can argue their own position with ever more force. It does tend to impede the debate.