Firstly can I say that I get where you're coming from, one of our children has severe respiratory issues, the first year had to be completely isolated at home (never went to baby group, met anyone except grandparents and our home nurse etc). RSV injections year 2 and 3, one lung as the other needed to be removed as it was so damaged, 8 pneumathorasis in the first 2 years, has had 3 heart surgeries, low Iga and MBL deficient, chronic lung disease, a mosaic form of a genetic condition affecting the lungs that is usually terminal within a year.... I get it basically.
With that said, what has their respiratory consultant advised? If it is complete isolation then that's what you need to do and in which case you're going to have to accept not living with a partner until your child has left home. We had to change jobs to work from home during the complete isolation period. If they're free to go to school, playgroup, a restaurant etc then there is no difference to your partner's child coming to your home. If you or their siblings go out to work and school then multiple people are going to bring back more bugs than one child? Are they on any treatment to mitigate the effect of common childhood 'bugs'? Our child has been on profilactic anti-biotics for 10 years now, as well as antihistamines to keep their lung dry as they have massive issues clearing mucus if they do get ill. However they also go to school, which despite also being a small class private the bugs don't seem to have a clue that I write a cheque for their education and they still come home coughing and spluttering every other week. So if they are going to school shortly try to look on the bright side and be glad of any immunity built up now as it's less missed school to come, it was really tough on ours to be out of school so much in the early days, but as time has gone on they are coping better with bugs and tend to only need 2 or 3 days off each time rather than 2 weeks and hospital which I think is just down to being that bit bigger. Without knowing the reason for your child's lung issues advice can be very different, however we were told when very small that lungs are very elastic, so the more they were used, the better condition they would be in, and advised 1-2 hours exercise every day. They found this very very hard at first, and we built it up, but now despite everything above they have a lung function capacity (depth of breath, control etc) better than 90% of children on most recent tests.
My thought is that you can't treat this child to a different set of rules than the rest of the family. Your house is their home just as much as their mums home. So while being younger they might be visibly ill a lot more than the adults, unless your child needs total isolation you can't really exclude them so reguarly. It's really, really scary knowing you have to expose your child to risk, but unfortunately for most of these children there is no way to mitigate it. If you really can't stand the idea of his child being around when ill then you need to permenantly live apart, not decamp if child has a bug, so that child still has two stable homes to go to.
I however agree with all of the posters saying paying for his child's private education is insane. Are you going to keep paying if you break up? That child has two parents who need to provide for them and make decisions in their best interest, but unless you're about to say your partner also chips in for your kids education so both are 50-50, or you won the euromillions so won't ever even notice that money, then this is quite possible the cheekiest of CF things I've ever read on mumsnet that they asked you to do this.