Significant others need to stay out of it. It will work better if the father of the child and the mother of the child work out these problems.
The father would probably benefit by brushing up on communication skills and learn to just focus on the issue at hand. Doesn't respond to his ex when she brings up any other subject. I find that often a minute of silence with no response eliminates or lessons the arguing. There is a web site that another mother has cited about non-confrontrational communication. This may help. If that poster doesn't send the address I think that you can find it by googling it with those words.
What may help is if and your significant other vent long before he makes the call. Then you support him in what he has to say in a positive way. It will help if he's thought of some specific non-confrontational ways to word his concerns while stating that he's not being critical but is wondering if there is some way that both families can prevent illnesses.
If they can't work it out, I agee that it may be best to just let it go if the illnesses aren't serious.
You didn't say in what way she is ill. You and your significant other may be able to figure some things out based on the symptoms and timing.
How long does the daughter stay with her mother? Is it possible that the daughter was exposed while at your house but didn't become ill until she was with her mother?
Does the other household also have small children? Does she go to daycare while with her mother? Are there other small children in your household? How is the health of the other children in her daycare?
Have you talked with her pediatrician to find out if she is ill significantly more often than the usual child her age. Have you taken her to the doctor to have any of her illness' diagnosed and documented?
What age is the daughter? If she's young, ie; baby, toddler, in preschool, kindergarten, 1st or 2nd grade I can tell you by experience that those ages have a tendency to come down with everything around. Their own immune system may not be fully functioning and they have lost most or all of the immunity that their mother had provided during pregnancy.
Also, there are new bugs coming around all the time. During my first year of teaching and my first few years as a police officer I was frequently ill with a fever and to the point of incapacitation. They were usually upper respiratory infections; more than a cold. After several years and being frutrated the doctor decided to remove my tonsils. This noticeably improved my health. There are several reasons that a child could be sick more often than another child would be. A new or different environment is the most common reason because the environment as bacteria and viruses that are also new. It takes time to find the cause(s).
Usually it is to a child's advantage to be ill in the early years because this helps their system to provide immunity for later years. Once their immunity is built up they have more left to fight the new bugs.
I have experience with my grandchildren and my daughter's grandchildren. They are freqently ill with a runny nose. The only reasons for exclusion from daycare and school are a fever of over 99 (I think that's the cut off)and must be fever free for 24 hours and have vomited in the last 24 hours or if they have a contagious rash as determined by the doctor or the school. They are also sometimes excluded if the mucous in thick and green or yellow.
My granddaughter has eczema and frequently has a rash that doesn't necessarily look like eczema. She's gone to school with it and no one questioned it. Pink eye is another concern. When my granddaughter's eyes looked infected the pediatrician said it was a response to allergies. She went to school. Perhaps the daycare as well as those involved in other activities would not be concerned if you had a diagnosis from the doctor.
You called them weird viruses. To me that indicates that the illness is different than that caused by an ordinary virus. If the virus is unusual than she does need to be seen by a doctor.
Allergies might be another possibility. She may be allergic to something in her mother's environment that you don't have in yours. Allergies also can weaken the immune system.
Who makes the decision about staying home? Since you haven't given her symptoms is it possible that you and her father are overly cautions? Perhaps the daycare would take her if they saw her?
You mention your community. I wonder what you mean when you say she cannot participate in some activities? What is their criteria? Is it reasonable given the activity?
Again, if the father and mother can't peacefully work it out I agree that it's best to just let it go. If the illness is serious or chronic then you do need to take his daughter to the doctor who can help you to determine if any neglect is going on. But then you would have to prove neglect which is tricky to do before you could possibly win in a custory hearing. Another reason to let go.
And the tension and fighting is hard on his daughter too. Anxiety can decrease the ability of the immune system to work. If she is hearing about this issue in both homes then both homes need to stop talking about it. You and your significant other can only control your own home. But you can influence her mother's home by not discussing it with her. Let the negative feelings die. It probably will take many months and maybe many years. The sooner you start the sooner the tension will ease.
Speaking of building up immunity, you could try giving her probiotics and vitamins. Does she have a healthy diet at your house? This should carry her thru at her mother's house. A good person to talk with concerning frequently illnes is a naturopath. They are more tuned into the environment and natural treatments.