Here is my take on the situation: if you were to mention to him that he is introducing the children to his new partners too quickly into the relationship, he will see it as being nosy and you trying to control what he does. In today's society, it appears that many of these people go through relationships like water and some are just a friends or friends w/benefits. These guys may not even realize what a real relationship is or even what love is, but I can see where he would get really bent out of shape if you were to suggest how he handles his relationships and getting his children involved. Trust me, if you were to point this out, he would get angry and not listen to a word you have said.
I know you are looking out for the best interests of your children, but you need to pick your battles, i.e., your child's asthma. I would send him a note and ask him if he's aware that your child's asthma can flare up if she is around cats and then you have to take her to the hospital for treatment. Ask him if it's possible to keep the cat in another room or elsewhere in the home while your daughter is visiting. Or, you could ask him if he has any suggestions on how best to deal with her health condition while visiting w/him.
For now, keep the focus on your child's health and what needs to be done to reduce the number of her asthma attacks.
Sit quietly, the answers will reveal themselves when you least expect them to. The past is gone, the present is a gift and you need to focus on today, allow the future to reveal itself when it is ready.
